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dry path
dry path
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@dry path Has your question been resolved?
dry path
Cmon
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frail blaze
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frail blaze
i got the other answer- just stuck on di)
i just dk like what way to solve it
smth like this?
wait
can i use the as y?
but thats not the inverse
is it?
wold it turn inverse if im using it for y
.close
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marsh ridge
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marsh ridge
I'm just confused what stuff like xRy means
old basin
It's translated to "x is related to y"
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@marsh ridge Has your question been resolved?
forest marsh
<@&268886789983436800>
pseudo chasm
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hearty quartz
yep derivatives
pseudo chasm
sketch 😮
hearty quartz
yes sketch
hearty quartz
What was the factorisation you got
hearty quartz
so you need to factorise the quadratic as well.
I was expecting you would perform the sketch using the factorisation
pseudo chasm
idk what to do with the info
hearty quartz
I will explain what to do. Just provide the factorisation
pseudo chasm
huh
hearty quartz
Just provide the expression
pseudo chasm
isnt the values of x this
hearty quartz
These are the exact values
pseudo chasm
so you need to factorise the quadratic as well.
yeah for the quadratic?
im lost
so x(x-3.19)(x+0.94)?
hearty quartz
I through you round numbers that why. It not big deal I guess
pseudo chasm
AH
hearty quartz
So we have the factorisation
hearty quartz
yep something seems off
pseudo chasm
pseudo chasm
range of x values just be clear
so the exact?
hearty quartz
We will just continue with expression we go for now. i.e.
so 4x(x-3.19)(x+0.94)
Now what happens when x<-0.94
Is the graph positive or negative
pseudo chasm
pos
pseudo chasm
so 4x(x-3.19)(x+0.94)
why 4x
hearty quartz
we had the 4 from 4x^2-9x-12
pseudo chasm
which graph are we talking abt
hearty quartz
Then 4x will be negative
pseudo chasm
derivative?
hearty quartz
Yes the derivative i.e. 4x(x-3.19)(x+0.94)
hearty quartz
Now what happens when x>0 and x<3.19
pseudo chasm
neg
hearty quartz
no
pseudo chasm
huh
pseudo chasm
both?
hearty quartz
Why would it be both
pseudo chasm
coz if x>0 neg but when x >2 its going up
the direction
or am i just looking at the a
if a then positive graph
hearty quartz
What happens if you multiply a positive number by positive number
pseudo chasm
how is that related
hearty quartz
It is related if x is positive then 4x is positive
if x is negative then 4x is negative
You seem to be over relying on desmos
pseudo chasm
so im not looking at the graph then? making me confused now
no because u said to graph
D:
hearty quartz
You should be able to sketch the graph roughly
i.e. know which parts of the graph are positive or negative
Let try again:
pseudo chasm
i see
pseudo chasm
pos
hearty quartz
much better
I am copying the expression for reference: 4x(x-3.19)(x+0.94)
Now is x-3.19 negative or positive
We assume x>0 and x<3.19
pseudo chasm
neg
pseudo chasm
pos
pseudo chasm
assume x>0 and x<3.19
why are we using this range btw
hearty quartz
I think I made a mistake earlier when I said no.
Since 4x is positive and (x+0.94) is positive and (x-3.19) is negative
We get a $positivepositivenegative = negative$
ocean sealBOT
team132
hearty quartz
In other words graph is negative for 0<x<3.19
hearty quartz
why are we using this range btw
The idea is consider all the possible cases and determine when graph is positive and negative
If graph is negative then that means derivative is negative hence the function is decreasing
If the derivative is positive then the function is increasing
Once x>3.19, (x-3.19) will be positive
Hence the function will be increasing for x>3.19
Similarly you can check other reigions
Interestingly the answer x>3,19 is not there. this suggest there is mistake in the question or made a calculation error earlier
pseudo chasm
Interestingly the answer x>3,19 is not there. this suggest there is mistake in t...
alright will ask the teacher. thanks a lot for being patient ❤️
hearty quartz
!done
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pseudo chasm
.close
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clever thunder
can i ask about q 10 i dont get it all😭
lone heartBOT
can i ask about q 10 i dont get it all😭
charred jewel
,rccw
ocean sealBOT
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@clever thunder Has your question been resolved?
exotic canopy
3 days learning
!status
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3 days learning
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
do you know how to find the domain and range of p?
clever thunder
1
clever thunder
do you know how to find the domain and range of p?
its okay man
i figured it out
.done
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wide glacier
For x,y,z>1 and (y+z-x-1)(x+z-y-1)(x+y-z-1) = 8 . Prove 1/x + 1/y +1/z <=1
cosmic nymph
have you started anything yet?
lone heartBOT
@wide glacier Has your question been resolved?
sharp panther
$\int{\frac{\floor{x}}{x^{cosec^{2}(x)}}dx}$
Can anyone give me a clue how to solve it
ocean sealBOT
@sharp panther
wind cloak
are there limits
ocean sealBOT
argon
sharp panther
are there limits
Nope no limits.
wind cloak
definitely isn't doable
sharp panther
definitely isn't doable
What should be the reason
wind cloak
,w integrate floor(x)/(x^(csc^2(x))
ocean sealBOT
sharp panther
,w integrate floor(x)/(x^(csc^2(x))
I wanna know the mathematical reason for it.
,w graph floor(x)/(x^(csc^2(x))
ocean sealBOT
wind cloak
there are proofs for integrals being non-elementary
and they're all out of my expertise
look them up yourself
lone heartBOT
@sharp panther Has your question been resolved?
fickle heath
Note that even x^x has no integral in terms of elementary functions, presumably neither do x^{1/x}, x^csc(x), or x^{csc(x)^2}
WA won't even give you an integral for floor(x) even though that's doable using floor... maybe floor isn't counted as standard
wet nest
There are two equation 1/4 (3x²+x-1) and x²-1
I know that roots of the first equation are between the roots of the the second equation
Then can i say that since dy/dx of 1 is 3/2x +1/4 and second is 2x i.e slope of dy/dx of 1> slope of dy/dx of 2 then both equation will intersect since rate of change of 1 is slower than 2?
lone heartBOT
There are two equation 1/4 (3x²+x-1) and x²-1
I know that roots of the first equ...
@wet nest Has your question been resolved?
neat ingot
yes... i think
i wouldnt say its because of "slope of dy/dx of 1> slope of dy/dx of 2" but more of 2x becomes strictly greater 3/2x +1/4, as x increases, and strictly less than 3/2x +1/4 as x decreases
but small details, its about the same
cinder breach
Is the closure of an open set always a closed set?
lone heartBOT
Is the closure of an open set always a closed set?
chrome lion
@neat ingot dms pls
half imp
Is the closure of an open set always a closed set?
the colsure is by defintion always closed
the closure of any set is closed
cinder breach
the closure of any set is closed
And the closure of a closed set Is the closed set itself?
half imp
yes
lone heartBOT
half imp
np
half imp
And the closure of a closed set Is the closed set itself?
this is iff btw
half imp
If set closure is itself, then it is closed
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opaque viper
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mossy laurel
Can you derivate $e^{-x^2}$?
ocean sealBOT
cjg#1618
mossy laurel
So can you integrate $xe^{-x^2}$?
ocean sealBOT
cjg#1618
opaque viper
I suppose by doing parts twice i could
mossy laurel
No you don't
opaque viper
But im not sure if im headed in the right direction
opaque viper
No you don't
Oh :/
mossy laurel
If the derivative of $e^{-x^2}$ is $-2xe^{-x^2}$, can't you integrate simply $xe^{-x^2}$?
ocean sealBOT
cjg#1618
opaque viper
Ohh i see ur point!
mossy laurel
Great
mossy laurel
You're welcome
opaque viper
.close
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dusty gyro
Hi, I'm coming with somewhat complicated problem (at least for me) that includes trigonometry and 3D spaces.
This will be a long read so I'm sorry about that. Also let me know if I should ask this somewhere else, thank you!
Context: I'm making a tool for a game called Teardown. The purpose of this tool is to allow player to move and rotate bones (points in 3D space) of a model so they can pose it or animate it (interpolation between two points in a 3D space). In order to achieve a proper behavior, I need to build a gizmo that will allow the player to rotate and move given point on a given axis. The rotation will be applied when player points their mouse cursor over given gizmo axis, clicks on it and drags the mouse while holding the mouse button. My issue starts here, as I need to account for player's cursor position, which is described in 2D as X and Y, where both of these variables are position on the screen in pixels. This will be better explained with pictures:
dusty gyro
I know this is in 2D, but it describes my general problem:
Point A is the position of the "bone" that will be rotated
Point B is the position of the player's camera (and the player themselves).
Point C is part of my current approach which is a point that moves correspondingly to the player's rotation
Let's assume that the dashed line represents current direction the player is facing, the issue is that point C does not perfectly move to intersect the dashed line.
What I tried currently is this:
pointA = Vec(x, y, z) ---- This is the position of the bone in 3D space
rotation = QuatEuler(cameraRotEulerX, cameraRotEulerY, 0) ----- This is to set dynamic rotation, this function returns Euler angles of the current rotation of camera in X and Y axes
mathSolution = VecAdd(GetCameraTransform().pos, QuatRotateVec(rotation, VecSub(pointA, GetCameraTransform().pos)))
-- ^ This is basically a function that will first substract current camera position in 3D space from pointA, then rotate the result by values in rotation, and finally add back the current camera's position
-- As a result, the point generally would rotate around the player as the player moves the mouse.
mathSolution[3] = pointA[3] -- this is to simply stop any movement in Z axis```Below you can find a recording of my issue in the actual game, I need the moving point to "stick" to player's cursor while respecting no movement across Z axis.
Let me add that I do not expect help with coding, I need to understand what math needs to be applied here so I can adjust the solution to fit in the code. Many thanks for all the help you can provide.
for more context, this is what would happen if I built the gizmo around Z axis with my current solution - red line is wrong, while green line correctly points towards player's cursor.
mental finch
Below you can find a recording of my issue in the actual game, I need the moving...
So if i understand correctly, you want the z position of the gizmo, to be fixed/locked, while the player is moving it?
dusty gyro
So if i understand correctly, you want the z position of the gizmo, to be fixed/...
Hi! I will show you exactly what I'm trying to replicate, one sec
fickle heath
It's vague, even with the video
mental finch
i appreciate that theyre alteast attempting to translate it for us and to something like desmos
fickle heath
Is your gizmo the representation of a 3D coordinate system (RGB axes) that is moving with the cursor but not exactly following it?
dusty gyro
Okay so this is a different tool that's doing what I'm trying to do
The reason why I don't just get the code from this is one that it uses some kind of system where a point is shot against a plane, I'm trying to figure out something simpler
In the video you can see the line pointing towards player's cursor, even if said cursor is no longer on the gizmo itself
mental finch
Hi, I'm coming with somewhat complicated problem (at least for me) that includes...
could you explain this desmos image again, and how that relates to this?
mental finch
The reason why I don't just get the code from this is one that it uses some kind...
oh youve got the idea figured out im guessing? maybe i missed it
fickle heath
I assume the Z axis is the one orthogonal to the camera plane? If so, I also assume that, during the interaction, you want to consider that Z axis at the moment you start the interaction (since it would change when the camera rotates)
dusty gyro
could you explain this desmos image again, and how that relates to this?
Sure!
So the desmos is a very simplified version of my problem, but in 2D. Basically the solution that I am trying to add is that there is a point in 3D space that follows mouse at all times. It needs to be be on the cursor at all times so I can apply math to determine the initial rotation (as 0,0,0 in Euler) and then add degrees correspondingly to how much the player has rotated the gizmo, just like in the second video.
Here's another video of my issue (I added big black box behind so the cursor is more visible) -- see how the point (represented by 3 axes lines) does not really stick to the white dot that is the player mouse? That's my issue because it would create wrong angle.
The vectors in the game are used also as a "locations" in 3D space, so for example something like Vec(0,3,4) would describe a position in a World space. I can also transform between World/local spaces (I suppose there is an equivalent of this operation in pure math, I think another coordinate system?) But this is not really important, because almost all numbers change - like bone position, player position, player rotation, player cursor.
The only static thing that I have is the player cursor position in 2D space, however the values for that are in pixels in relation to screen size, so not really applicable in 3D.
However! I do have a function that "translates" cursor position to 3D space, but it returns a normalized 3D vector describing the direction the player is looking at (you know, for example facing "East" world direction, the values go below 0 on X value to max -1 which would be facing exactly East)
I am willing to provide as much info as you need guys, I am trying to solve this for days and I know I'm getting close.
If you look closely you will also see the red line that goes from the "bone" to the moving point, and you can tell it's not pointing towards the cursor.
in relation to desmos, I want this, at all times, in 3D.
I can tell that the point C needs to move across "Z" plane somehow. Oh also, camera rotation can be both expressed in quaternion and Euler angles if that helps.
fickle heath
I still don't understand where you want to gizmo to be exactly (all three coordinates)
dusty gyro
I still don't understand where you want to gizmo to be exactly (all three coordi...
Oh, to rotate the bone? Well, on the bone position. I already have this covered, the gizmo is not really part of the problem - it wil always be located on one of the bones.
That moving point is just to determine the rotation on the gizmo itself
for example:
Pink line would be pointing towards that moving point, and the point itself is "glued" to the mouse cursor, so in the end pink line points towards mouse cursor. (Red circle in the picture)
fickle heath
So what are you trying to find? The coordinates of the pink line or of the point that is glued to the cursor? Both?
dusty gyro
A correct answer to that would be the position of the cursor in 3D space, on the Z axis (I'll copy the solution to other axes later)
Then, once I have that, the point will move with the cursor and I can build a relation between the cursors position and the bone position.
(the video shows the different tool I want to replicate that respects player's cursor position at all times, no matter where the player stands)
mental finch
could you also possibly overlay your current solution, compared to this one? to get a better sense of the issue
dusty gyro
Also guys I want you to know it's totally okay if it's something hard to imagine or not the right discord to ask about, I'm just trying everything at this point and I appreciate all help <3
Sure Aslan, one sec!
mental finch
Also guys I want you to know it's totally okay if it's something hard to imagine...
might be worth mentioning that there's an okay-ish not so active channel in a coding server which deals with stuff like this!
a channel way down called #math- ... something
dusty gyro
Oh, thank you, I'll take a look there as well in a sec!
fickle heath
The reason why I don't just get the code from this is one that it uses some kind...
Not sure what you mean by "a point shot against a plane", but if it's just a projection, I doubt you'll find simpler
mental finch
also from what ive understood so far, this seems like a somehwat subjective thing to decide how to do
fickle heath
Basically I would project the bone position (gizmo center) onto the camera plane, draw a line in that plane between it and the cursor, then reverse the projection
dusty gyro
Okay so, other tool vs my tool
Additionally, for example in my tool this is wrong:
The green line represents where I want the point to be
dusty gyro
also from what ive understood so far, this seems like a somehwat subjective thin...
Totally! The whole issue is that the game doesn't have anything I can use to translate 3D point onto the middle of the screen while restricting movement on one of the axes, that's why I'm figuring it manually
dusty gyro
Basically I would project the bone position (gizmo center) onto the camera plane...
This sounds doable, I might need to look deeper into projection then and somehow translate math --> code.
fickle heath
I may have spoken too fast, "reversing the projection" isn't quite right
But also isn't this just a line-plane intersection problem?
dusty gyro
If we think about using plane as a solution, then yes -- I project "some" point onto the plane and need to figure out where it actually lands.
fickle heath
So you want the "point that is glued to the cursor" (you should give it a name) to stay in some plane in 3D that has nothing to do with the camera, and where exactly in that plane it is depends on where the cursor is?
mental finch
also if it hasnt already been mentioned, what useful information do you already have given?
dusty gyro
So you want the "point that is glued to the cursor" (you should give it a name) ...
Let's name it just point A,
So, let's imagine this, I assume the plane could also be in the position of the bone, like in the picture. Now the red circle would be a point that I'm casting from the camera position (so technically it always stays in the middle of the screen)
dusty gyro
also if it hasnt already been mentioned, what useful information do you already ...
Sorry, can you explain this question?
Ah, sorry, the picture is wrong, jesus, not "Z", but "X"
fickle heath
Let's name it just point A,
So, let's imagine this, I *assume* the plane could ...
Not sure what those axes represent
mental finch
like, what type of information from the game do you have access to; which is useful to know when solving the current problem
fickle heath
Isn't the plane that we are talking about the plane of rotation?
mental finch
are we working only from one given point? etc?
mental finch
Isn't the plane that we are talking about the plane of rotation?
it seems so
fickle heath
From what I understand you want the interaction to rotate some object in 3D, which means you have a plane of rotation
If all you want is to place your point A in that plane but also in the center of the screen from the camera 's perspective, then all you need is a line-plane intersection
mental finch
and there's most likey not a shortcut "simpler" way of doing it, its just basic linear algebra; unless im missing something
dusty gyro
like, what type of information from the game do you have access to; which is use...
Oh! So, I can have any World position I want, I have transform of the camera (both position and rotation in quat/Euler if needed), I have direction I am looking at as a 3D Vector (from the player's "head" position you can say), I have mouse position in 2D space, any bone position in 3D space (it can also be translated to pixels/screen position but I don't think it would help). Basically if anything exists on the screen, I can get info on where it is, what is it's rotation and other game-related info.
dusty gyro
If all you want is to place your point A in that plane but also in the center of...
I see. Okay, so there's no easier way to do this, got it!
fickle heath
I have mouse position in 2D space
Isn't the mouse always in the center of the screen though?
dusty gyro
> I have mouse position in 2D space
Isn't the mouse always in the center of the ...
yeaah but you can disable that and have a moving cursor while the camera is stationary, then the numbers are different, not really relevant to this I think
mental finch
so is that somehting youre taking into account aswell? if a player decides to do that(?)
fickle heath
yeaah but you can disable that and have a moving cursor while the camera is stat...
It seems pretty relevant
dusty gyro
Nope, that will just be ignored, I will disable that input and check if some variable for that is either true or false
mental finch
Oh
dusty gyro
This is more of a design choice decision, do I want player to move the gizmo without moving the camera or not, which is most cases the answer is no
It's just easier to see what's going on when you move the entire camera
mental finch
can the other tool do that?
Hm also, i guess if you dont happen to find an easier method. Doesnt that mean you can fall back to whatever their tool does?
dusty gyro
can the other tool do that?
nope, they also block the input from "free mouse"
mental finch
I see!
dusty gyro
Hm also, i guess if you dont happen to find an easier method. Doesnt that mean y...
That was my first instincts - just check their code but holy hell it's a mess, that's why I started wondering in a first place.
mental finch
Ah
dusty gyro
I probably still will do the same as the author did, but more... tidy
mental finch
as soon as you hear line or plane, there's most likley a really neat way of doing it trhough the perspective of linear algbera
dusty gyro
Anyhow, it's seems that the only solution here is to project the point onto the plane, so I guess I have my answer :D
I just need to figure out how to determine the plane in the first place, and... well... Cast the point, it might not be that easy as in regular math :D
Thank you guys so much for working with me on this, onto the drawing board I go!
mental finch
also again worth taking a look into that coding server!
fickle heath
That was my first instincts - just check their code but holy hell it's a mess, t...
the joys of working on someone else's code
dusty gyro
oh yeah, almost forgot
mental finch
good luck!
dusty gyro
<:KEK:586240877358350341> the joys of working on someone else's code
yeah well I---
500 lines.
of this.
so yeah :D Thanks guys again, I'll close this one out B)
fickle heath
Anyhow, it's seems that the only solution here is to project the point onto the ...
It's not a projection, it's a line-plane intersection
A projection is technically a line-plane intersection but the line would be orthogonal
dusty gyro
ohh okay, yeah that's probably crucial, thank you
Aight, gotta work! Have a great day amigos!
.close
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bronze eagle
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bronze eagle
Can someone please explain why poo and paa = 1? Or is it just the definition of these kinds of matrices?
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<@&286206848099549185> kindly help.
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
vale crag
Can someone please explain why poo and paa = 1? Or is it just the definition of ...
they explain just after the signification of p00 = paa = 1
if one of the two players is ruined, there's no game that can happen anymore
if the chain is at state 0, the gambler is ruined and will always stay ruined
if it's at state a, the adversary is ruined and will always stay ruined
vale crag
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maiden bough
Solving 2x-25=8 for x. I am taking my intro chem class and would like some help to find the solution
mellow bolt
Solving 2x-25=8 for x. I am taking my intro chem class and would like some help...
when solving an equation like this, we always need to do the same thing to both sides
and our end goal is to find out what x is
so the first thing i wanna get rid of on the left hand side is the -25, so i add 25 to both sides
2x - 25 + 25 = 8 + 25
that gets 2x = 33
then i wanna get rid of the 2 that has been multiplied by x on the left hand side, so i divide by 2 on both sides
2x ÷ 2 = 33/2
so x = 33/2
maiden bough
oh thank you! I understand it now 😄 much appreciate it!
mellow bolt
nws
wicked carbon
lone heartBOT
wicked carbon
This is the work I have done so far
I am trying to find the foci, but I'm unsure of how to handle this -12, all previous answers have been positive
what I'm seeing online says if this is negative then it's a circle, when I graph the equation on desmos it gives an ellipse so this is conflicting information and I'm confused
also i think I made a mistake and this is vertical not horizontal because the y denominator is larger than the x denominator
lone heartBOT
@wicked carbon Has your question been resolved?
lone heartBOT
@wicked carbon Has your question been resolved?
small stag
it is a horizontal ellipse
Your mistake is equating "a" to the number under the x^2 term
really "a" is (half) the length of the major axis, which is in the y-direction in this case
one way to easily tell which way the ellipse is pointing without having to use some memorization is to look at your equation
looking solely at the $\frac{x^2}{4}$ and $\frac{(y-1)^2}{16}$ terms individually
ocean sealBOT
vehnil
small stag
I ask myself how big x^2 and (y-1)^2 need to be respectively for the output to still = 1
you can see that the $(y-1)^2$ in $\frac{(y-1)^2}{16}$ needs to have a much bigger magnitude at its vertices than $x^2$ does at its vertices
ocean sealBOT
vehnil
small stag
that tells you its stretched in the y-direction, so major axis is in the y-direction meaning the foci (which always lie on the major axis) are also oriented on the y-direction
you can find the foci using the pythogorean theorem because focis have a special right triangle relationship with the half-lengths of the axis (a and b)
frozen jasper
lone heartBOT
frozen jasper
How do I figure out this question?
small stag
the first average is just a sum of the first four numbers divided by four
which means the sum of the first four numbers is 92 * 4, or 368
You know the last number in the original list is 40
so subtracting that, you get the sum of the first three numbers 368-40 = 328
cinder tundra
!nosols
lone heartBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
small stag
Using this you can find the new average
frozen jasper
I see, thank you for your help @small stag ❤️
small stag
Another way to think about it is seeing the fourth number is increasing by 8
which means the average will increase proportionally
frozen jasper
So adding 8 and finding the average that way?
small stag
Think about it this way
say I have this set of numbers: 5, 5, 5, 5, 5
Its average is just 5
frozen jasper
yes
small stag
so its like $\frac{5+5+5+5+5}{5}$
ocean sealBOT
vehnil
small stag
now instead if I add 5 to the last number my set is now 5, 5, 5, 5, 10
do the average out on your own and tell me what you get
frozen jasper
6
small stag
lets try to figure out the idea behind how to do a problem like this when you don't know the specific numbers
using the first way I said
frozen jasper
you can do this with variables?
small stag
I could but its gonna be easier to use real numbers rn and then convince yourself with variables after
We know that we have a list of 5 numbers and their average is 5
actually lets use variables now, I think that might be easier
so with the list $ x_1,x_ 2, x_3, x_4, x_5$, the average is $\frac{x_1+x_2+x_3+x_4+x_5}{5}$
ocean sealBOT
vehnil
small stag
We can find the sum of all the variables $x_1+x_2+x_3+x_4+x_5$ by just multiplying the average by the number of numbers (in our case 5)
ocean sealBOT
vehnil
small stag
x_1+x_2+x_3+x_4+x_5 = 5*5 = 25
then when we want to find the average when one of the numbers change, we can just subtract what that number used to be and add what it is now
x_1+x_2+x_3+x_4 = 25 - 5 = 20
x_1+x_2+x_3+x_4 + 10 = 20 + 10 = 30
so the new average is 30/5 = 6
frozen jasper
ahh I see. because we know that our sum must be 25
small stag
exactly
the second way I was talking about isn't really necessary but it can be a good way to view the problem
Instead of worrying about the actual sum you can just think about how much the average changes proportionally
Say $y_5$ is the new last number and we know that $y_5 = x_5 + 5$
ocean sealBOT
vehnil
small stag
this is just me writing 10 = 5 + 5 where y_5 = 10 (new last number) and x_5 = 5 (old last number)
does that make sense
frozen jasper
yes that does make sense 👍
small stag
now we know the new average is $\frac{x_1+x_2+x_3+x_4+y_5}{5}$
ocean sealBOT
vehnil
small stag
Say $y_5$ is the new last number and we know that $y_5 = x_5 + 5$
but since we know this we can think about this new average as actually like
$\frac{x_1+x_2+x_3+x_4+(x_5 + 5)}{5}$
ocean sealBOT
vehnil
small stag
if we split some stuff up we get $\frac{x_1+x_2+x_3+x_4+x_5}{5} + \frac{5}{5} = 5 + 1$
ocean sealBOT
vehnil
small stag
Exactly
So all of this is just the algebraic way of a much simple concept
Basically if you increase/decrease one of the numbers in a set of n numbers by some amount K
the average will increase/decrease proportionally by K/n
because the average is just a scaled down sum
frozen jasper
incredibly well put @small stag
small stag
Sometimes it can make problems like these solvable in like 5 seconds
you just need to compute the change and in which direction
small stag
but its not necessary, you can always do it the old fashioned and its always going to work
i see i see
frozen jasper
I have one last question that I am currently struggling with. I think the word questions confuse me slightly. Do you know how to solve this problem?
small stag
since the living room and the tiles are measured in different units, to make this easier for yourself you will have to convert one of those units into the other
either make the tiles into feet or the floor dimensions into inches
I would do the latter but whichever seems easier to oyu
frozen jasper
2.77 feet per carpet tile. 8 1/3 * 10 = 83.33
83.33/2.77 = 30.08
Which will give me D
thank you for the tip of converting. I think I figured it out!
small stag
how many inches in a foot
frozen jasper
12
frozen jasper
20/12 = 1.66
small stag
I meant more like
small stag
so the room is 8 1/3 by 10 feet, make those into inches and its (96 + 4) by 120 inches
or 100 by 120 inches
and that way you can see its 5 tiles by 6 tiles
like you can fit 5 tiles on one edge and 6 on another
the thing with these problems is that sometimes it won't be even
Like you'll have some excess on the sides
so you round up
so say it was like 5.75 by 6.75 tiles
you round up to 6 * 7 = 42
but if you purely divide by area you would get like
38 or 39
frozen jasper
Ah so do you think that my method is taking some liberties?
small stag
But I don't think they will make you do that on the act tbh
slightly but it probably doesn't matter
frozen jasper
have you taken it before?
small stag
its kinda like assuming you will use the scraps of some material you are using
say there was a 2 inch excess, no one is really gonna fill in the edges with 2 inch wide carpet
No I took the SAT 2 years ago but I did do some ACT practice
frozen jasper
say there was a 2 inch excess, no one is really gonna fill in the edges with 2 i...
yes exactly
frozen jasper
No I took the SAT 2 years ago but I did do some ACT practice
Do you have any tips for the mathmatics section of either test?
small stag
I think speed is important but don't obsess over it
Technically its like what 1-2 minutes per question but thats an average
its designed so that some questions take 10 seconds so that others take much longer
I would say that certain books do cover the chapters well
I didn't ever study for math but I did use the SAT black book for english and I heard its math section is good too
There is an ACT version too
very good book and you can download it online too
But if there was any book I would recommend paying for, its this one
and I really don't like books so that means something
frozen jasper
wow. Im not the greatest fan of books myself, however I struggle heavly on the math section vs the english and science portions
so I am 100 precent willing to study with any material LOL
small stag
yeah I get it
frozen jasper
I have a greater understanding of those questions though, I appreciate it man.
frozen jasper
Well, im taking it tomorrow (I know, yikes) and then im taking it again in september, and one more time in october.
small stag
You'll do great
Sleep is more important at this point than studying imo its a long test
how long is act
frozen jasper
right. 4 hours
small stag
yeah you can't get by with not enough sleep
frozen jasper
2 breaks
small stag
Eat a good meal tonight, sleep early, wake up early and maybe a quick snack in the morning
No screens for at least an hour before bed, I even went as far as leaving them outside of my room
And if it seems like a lot, just remember its just one night in the grand scheme of things doing all this isn't a big deal
so you should do them
If you really want, do like 10-15 practice problems on the drive to your testing center to prime your mind
frozen jasper
yes. Anything to optimize my chances
small stag
are you applying to college this year or next
small stag
home stretch
frozen jasper
yes
small stag
yeah
small stag
you got it
frozen jasper
so im attempting act
my math is terrible though
but ive got a little time
im going to work for a little more tonight
and fix it
small stag
I would recommend the black book for the next two tests if you do plan on taking them
but not for tonight its too much volume
you can dm me for questions if you want, I can explain the way I think about them
I turn of notis for this server so just dm actually
frozen jasper
absolutely. There are some quadtratic trig questions that kill me
I will DM you
thank you again @small stag
small stag
of course!
lone heartBOT
@frozen jasper Has your question been resolved?
silent slate
How do I put 0+ in a TI-84 plus calculator without an error message showing
silent slate
How do I put 0+ in a TI-84 plus calculator without an error message showing
To anyone that can help, please @ me when you can respond
ebon sparrow
How do I put 0+ in a TI-84 plus calculator without an error message showing
are you trying to write a limit or smth
silent slate
why not just 0
I put that in but my calculator was giving an incorrect answer
silent slate
I feel like there's something missing here
I just fiddled with my calculator some and found the issue, I'm sorry for wasting any of your time
I think typing it out helped though, thank you both
mental finch
I think typing it out helped though, thank you both
Was it related to how the floating point numbers -0 and 0 are sometimes treated differently?
silent slate
Was it related to how the floating point numbers -0 and 0 are sometimes treated ...
It was the lists that were set as frequency or X, I had L1 and L2 swapped by mistake
I did that just the question before so it had me confused
frozen jasper
lone heartBOT
frozen jasper
Is there anyway I could have help with thsi question?
tawdry urchin
multiply the numerator and denominator of each fraction by an expression to make the denominators the same
frozen jasper
So like mutply them each by (x+y)
frozen jasper
do I mutiply the fraction x/x+y by (x-y) in the numerator and denominator?
and multiplyu the fractoin y/x-y by (x+y) in the numerator and denominator?
small stag
yeah
lone heartBOT
@frozen jasper Has your question been resolved?
spark tartan
If my book says to round using significant figures, but it doesn't tell me which significant figure to round to, how do I know what to do? Also if all the zeros after a decimal point after the other numbers are significant, wouldn't there be an infinite number of significant figures?
versed birch
If my book says to round using significant figures, but it doesn't tell me which...
does it specify how many significant figures
spark tartan
does it specify how many significant figures
No it doesn't
versed birch
yeah if it doesn't really specify you can't really tell
spark tartan
It just says use proper significant figures
half imp
it is based on the numbers in the problem
lone heartBOT
@spark tartan Has your question been resolved?
fresh yarrow
If my book says to round using significant figures, but it doesn't tell me which...
usually whichever given value has the least number of sig figs will be what you round to
so if it gives you 10. and 1123.412, you round to two
also the zeros after a decimal point are only significant if there are nonzeros after
ie 4.0002 has 5 sig figs
sometimes books do weird things like 300.0 and want you to read it as 4 sig figs, but its not super common ime
spark tartan
usually whichever given value has the least number of sig figs will be what you ...
My book says they are significant even if it's all zeros at the end explicitly
And I got super confused
fresh yarrow
My book says they are significant even if it's all zeros at the end explicitly
yea, it be weird sometimes but thats a common book thing. i think they just mean that read all digits given as significant
spark tartan
yea, it be weird sometimes but thats a common book thing. i think they just mean...
What if it's on a calculator and it's all 0s do I write them all down?
fresh yarrow
What if it's on a calculator and it's all 0s do I write them all down?
nah you make your answers the same number of sig figs as what the book gave you
spark tartan
nah you make your answers the same number of sig figs as what the book gave you
I don't get it. It's not giving me the numbers the calculator is
Can you vc in a few minutes
half imp
the calculator will not tell you how many sig figs you have
you need to learn the rules yourself
fresh yarrow
Can you vc in a few minutes
ye prolly
spark tartan
the calculator will not tell you how many sig figs you have
I know
fresh yarrow
What time. I need at least 15 minutes
im free, just ping me
spark tartan
im free, just ping me
Alright
spark tartan
im free, just ping me
where is the vc?
fresh yarrow
where is the vc?
lowkey idk, they might even have one
spark tartan
lowkey idk, they might even have one
can i copy paste what my book says here?
fresh yarrow
ye
spark tartan
Significant figures
Whenever you communicate scientific calculations, you need to ensure that you use proper significant figures (which are sometimes called significant digits).
A significant digit (also called a significant figure) is one that is actually measured. The number of significant digits in a measurement depends on the measuring device and the magnitude of its smallest graduation (for example, on most rulers, the smallest graduation is a millimetre).
Determining the number of significant figures in a measured quantity requires a few rules. The difficulty occurs primarily when there are zeros. Explore the following chart to know the rules.
Rule applied Examples
All digits 1-9 (non-zeros) are significant. 193 has 3 significant figures.
Zeros between significant digits are significant. 6003 has 4 significant figures.
Leading zeros (those with no significant digits to the left) are not significant. 0.00345 has only 3 significant figures.
Trailing zeros (those to the right of all other significant digits) are significant only if a decimal point is present. 0.000000980 has 3 significant figures.
0.14000 has 5 significant figures.
It is important to be accurate in both the taking and reporting of your measurements. The accuracy of a measurement/numerical value/number can be determined by the number of significant figures it comprises.
When performing calculations, after calculating a final answer, you must use significant figures to report your conclusion. In general for this course, your final answer should have the same number of significant figures as the smallest number of significant figures used in the calculation. This limitation only affects the communication of your value. If you use that same number for future calculations, you should use the full un-rounded value, then round again when reporting the next answer.
just realize I have perms
fresh yarrow
which part are you stuck on?
spark tartan
which part are you stuck on?
how do I know which significant figure to round to
none of these questions specify
fresh yarrow
how do I know which significant figure to round to
you should use the full un-rounded value, then round again when reporting the next answer.
its kind of this part
spark tartan
any why are all those zeros significant
fresh yarrow
and thats again just the book being pedantic, the book gives you some values, you do the math, and then when youre turning in your answer you round your final values to whichever lowest sig fig was used from the original problem
spark tartan
> you should use the full un-rounded value, then round again when reporting the ...
okay but when I round to report my answer how can I do that?
when a number is 0.0045 why aren't those zeros significant?
fresh yarrow
okay but when I round to report my answer how can I do that?
for example,
calc Area of a circle with a radius 3.15
we do the math the area is 31.17245...
but we only had 3 sig figs from the problem 3.15
so our final answer is 31.2
fresh yarrow
when a number is 0.0045 why aren't those zeros significant?
its easier to see in sci notation if you know that
0.0045 = 4.5 *10^-3
but 0.00450 = 4.50 *10^-3
spark tartan
for example,
> calc Area of a circle with a radius 3.15
we do the math the area...
so say the only number given is 98.0, would I round to the second significant figure for all my answers in that question
fresh yarrow
so say the only number given is 98.0, would I round to the second significant fi...
the 3rd, cause if it was only 2 sig figs it would be 98.
but yeah
spark tartan
isn't 98 and 98.0 the exact same number
fresh yarrow
in pure math, sure
in science and stuff, not really
spark tartan
whats the difference in science
fresh yarrow
if i have a scale that can measure accurately only to 1/10th of a lbs, and i weigh something thats 12.34lbs my scale will only read 12.3lbs. i cant be certain that the last digit is any number, so the last siginificant digit (the ones i can read) would be the .3
spark tartan
I randomly panic when im doing math so sorry if I take 5 seconds or start saying stuff that are incoherent
fresh yarrow
its chill, man, no worries
but the scale thing is why 0.005 has sig figs that are zero, im saying i can accurately measure the 5/1000ths so i can confirm the 100ths and 10s places are truly zero
spark tartan
if i have a scale that can measure accurately only to 1/10th of a lbs, and i wei...
okay so are you saying 98.0 could be 98.00000001 or something
spark tartan
but the scale thing is why 0.005 has sig figs that are zero, im saying i can acc...
the book seems to say there is only one significant figure in 0.005
fresh yarrow
the book seems to say there is only one significant figure in 0.005
yeah wait i messed that up
i meant if my scale reads 5.000 i know those are all zeros but there could be some stuff after
so like 98.0 could be 98.0112354 but i cant read it so only the first 3 are significant
the main thing i want to say is if your book writes a number weird (like saying 98.0 instead of 98) theyre doing it because that 0 is significant and should be included
spark tartan
the main thing i want to say is if your book writes a number weird (like saying ...
so that would be 3 significant figures?
if they put that
for the whole question?
fresh yarrow
if theres no other values given, then yes
spark tartan
okay but what if there are more than one value given
im completely lost on this
everything else is fine up to this simple point in the first unit that I skipped over
fresh yarrow
okay but what if there are more than one value given
if i have multiple values i pick the least number of sig figs to round to
ie if i have 9.811, 10.9 , 123.54,
which are 4 3, and 5 sig figs each
i should round to 3 sig figs (from 10.9 having 3 sig figs which is the least amount of sig figs)
spark tartan
if i have multiple values i pick the least number of sig figs to round to
ie if ...
what about a number like 404.95867768595
all the numbers seem significant
fresh yarrow
is that given to you by the book?
spark tartan
it was one of my calculations
fresh yarrow
then its sigfigs are not relevant
spark tartan
why not
it says to do this for calculations
im going back into panic, idk if I should even be saying this on here
fresh yarrow
im going back into panic, idk if I should even be saying this on here
first of all, deep breath chill for a second
but significant figures are only important for reporting data. the book reports data to you, you then report something back, so the two times we need to care about sig figs is how many we get = how many we send back
spark tartan
but significant figures are only important for reporting data. the book reports ...
it says final answer and this is the final answer for this question so it has to be rounded for significant figures
fresh yarrow
it says final answer and this is the final answer for this question so it has to...
okay good, so we just need to round it to least accurate sig figs we used to calculate it.
spark tartan
okay good, so we just need to round it to least accurate sig figs we used to cal...
what does that even mean
fresh yarrow
what does that even mean
the less sig figs the less accurate the number
what do you think you should round this 404.95867768595 to?
spark tartan
404.96
fresh yarrow
and the question had no values with less than 5 sig figs right?
can you send the question maybe?
spark tartan
fresh yarrow
so we have two values, 98.0 and 0.00242
they have sigfigs of 3 and 3
so we need to round our answers to 3 sigfigs
spark tartan
so we have two values, 98.0 and 0.00242
they have sigfigs of 3 and 3
so we need ...
do I only have to round decimals? What about when the significant figures are before the decimal are those rounded too
like the number 404.9587
wouldn't the 3rd sig figure be 4
that makes no sense
fresh yarrow
wouldn't the 3rd sig figure be 4
what do you mean?
spark tartan
So far in the answer keys they only round the number on the other side of the decimal
The after the decimal
fresh yarrow
so they round to 405.0?
spark tartan
No no no
I don't know I might have to take a break
Will it close the ticket
I want it open for while
fresh yarrow
if theres no activity for a while it does time out eventually
spark tartan
if theres no activity for a while it does time out eventually
Doesn't it ask me if I want to close it
fresh yarrow
Doesn't it ask me if I want to close it
im not sure
lone heartBOT
@spark tartan Has your question been resolved?
spark tartan
<@770427154676318251> Has your question been resolved?
no wtf
lone heartBOT
@spark tartan Has your question been resolved?
lone heartBOT
@spark tartan Has your question been resolved?
spark tartan
what's your question?
im not feeling good just give me a second
twin nimbus
@spark tartan
How to figure out the number of significant figures to use:
During a multiplication or division: use the smallest number of available significant figures. So if you are multiplying a number with 6 significant figures with a number with 3, then you use 3 significant figures for your result. This doesn't include exact values. (i.e. if you have exactly 2 of an item that each weighs 4.56 g you report the answer as 9.12 g not 9 g.)
When adding or subtracting, use the level of significant figure corresponding to whichever least significant figure has the higher exponent. A little bit confusing, but if you have:
43.2 - 39.8374 you report only 3.4 because the least significant placement of digits is the tenths. Note this means that adding can actually produce additional significant figures, and subtraction can destroy them.
lone heartBOT
@spark tartan Has your question been resolved?
hearty badge
Maybe say that e.g. $E_{ij}$ denotes the Matrix of the desired dimension with zeros in every slot but the ij one
And write the values of i,j for which each E is in the basis
ocean sealBOT
Ij37
lone heartBOT
Available help channel!
thick lynx
lone heartBOT
keen plinth
this is not literally what U_1 + U_2 + U_3 is
,align U_1 + U_2 + U_3 & = \set {\mat {x \ y \ 0} + \mat {0 \ 0 \ z} + \mat {0 \ w \ w} \where x, y, z, w \in K} \ & = \set {\mat {x \ y + w \ z + w} \where x, y, z, w \in K}
this would be more appropriate
ocean sealBOT
thick lynx
,align U_1 + U_2 + U_3 & = \set {\mat {x \\ y \\ 0} + \mat {0 \\ 0 \\ z} + \mat ...
Ok, thanks. And otherwise, you would do it like this too? Show that in its span, it's just the canonical basis of K^3?
keen plinth
in its span, it's just the canonical basis of K^3
what?
if (1, 0, 0), (0, 1, 0), (0, 0, 1) are contained in U_1 + U_2 + U_3, then U_1 + U_2 + U_3 will be K^3
thick lynx
if (1, 0, 0), (0, 1, 0), (0, 0, 1) are contained in U_1 + U_2 + U_3, then U_1 + ...
So you'd just show that those are contained in this set?
keen plinth
well its sufficient
thick lynx
Ok, thanks!
thick lynx
well its sufficient
Btw how do you do the large $\mid$
ocean sealBOT
Kepe
keen plinth
,, \left(\int\middle|\int\right)
ocean sealBOT
lone heartBOT
worldly surge
For a given positive integer m, you have a round table of 2m seats, and m couples attending this party (all 2m people are distinct). Given an odd integer
1 < n < m, you want to seat everyone such that the two people in each couple
are exactly n seats apart (as in there are n−1 seats between them). How many
distinct ways are there to achieve this?
help with this please
worldly surge
can you help with this
worldly surge
thnx cutiepie 😭😂
thick lynx
.reopen
lone heartBOT
✅
thick lynx
,, \left(\int\middle|\int\right)
\begin{align*}
U_1 + U_2 + U_3 &= \l{\mrm{x \ y \ 0} + \mrm{0 \ 0 \ z} + \mrm{0 \ w \ w} \mid x, y, z, w \in K\r}
\end{align*}
Now $\mrm{1 \ 0 \ 0} \in U_1 + U_2 + U_3$ with $x = 1$ and $y = z = w = 0$. $\mrm{0 \ 1 \ 0} \in U_1 + U_2 + U_3$ with $y = 1$, $x = z = w = 0$ and $\mrm{0 \ 0 \ 1} \in U_1 + U_2 + U_3$ mit $x = y = w = 0$ und $z = 1$. Because of $\l \langle \mrm{1 \ 0 \ 0}, \mrm{0 \ 1 \ 0}, \mrm{0 \ 0 \ 1} \r \rangle = K^3$ ist $K^3 \subseteq U_1 + U_2 + U_3$, but also $U_1 + U_2 + U_3 \subseteq K^3$ since $U_i$ are subspaces of $K^3$; thus $K^3 = U_1 + U_2 + U_3$.
Does it look fine like this?
worldly surge
yea
keen plinth
yeah this is fine
thick lynx
Thank you!
worldly surge
yea :shrug:
you can help me in my own help channel 38
thick lynx
yeah this is fine
Now if we want to know if $K^3 = U_1 \oplus U_2 \oplus U_3$ or not, how do we argue there?
keen plinth
well dim U_1 = 2, dim U_2 = 1, dim U_3 = 1, so it cant be a direct sum
thick lynx
well dim U_1 = 2, dim U_2 = 1, dim U_3 = 1, so it cant be a direct sum
This already fulfills the first requirement, K^3 = U_1 + U_2 + U_3
If every v in V can be uniquely written as v = u1 + u2 + u3 we are done and this is actually U_1 oplus U_2 oplus U_3
You argue differently?
keen plinth
,, \dim (U \oplus V \oplus W) = \dim U + \dim V + \dim W
ocean sealBOT
thick lynx
Oh
keen plinth
but if you want to argue your way
you can see that $U_1 + U_2 \supsete U_3$ in your example
ocean sealBOT
thick lynx
Thank you!
keen plinth
👍
thick lynx
👍
For a) it's fine if I just say e.g. (x, y, 0) = (0, 0, z) iff z = 0 and x = 0, y = 0 thus the only element is 0_3, etc.
Right?
Comparing the components
keen plinth
yeah
lone heartBOT
lone heartBOT
Available help channel!
uncut tide
hey i need help with part ii)
lone heartBOT
hey i need help with part ii)
uncut tide
this is what i am thinking of doing
but i dont know where to go from here
does it maybe want me to sketch a graph?
mental finch
So if possible do that, I would also try and see if a cheap trick like seeing if the LHS is strictly increasing for certain k is possible here
And if you can show that the LHS exhibits both positive and negative values, then you know there’s always a unique root for those k where it is strictly increasing
ocean sealBOT
uncut tide
here are some rough notes, i think i kind of getting it
if we add 1 to the karctan(phi) it will move it up 1
and then the e^phi and the karctan(phi)+1 will intersect
alpine sable
Consider the arithmetic sequence 2 ,12 , 22 ,....... . .
a) What is the algebraic form of this sequence ?
anyone
alpine sable
<@&286206848099549185>
small lance
thick lynx
\begin{problem} Let $V$ be a $K$-vector space and $U_1, U_2, W \subseteq V$ subspaces. Is the following proposition true or false: $V=U_1\oplus W=U_2\oplus W \implies U_1=U_2$. (proof or counterexample.)
\end{problem} I feel like this is true, am I missing a counterexample or is it actually true?
ocean sealBOT
Kepe
thick lynx
do you think it holds in R²?
$U_1 \oplus W = W \oplus U_1$, right?
ocean sealBOT
Kepe
thick lynx
So this should be false..
marsh rapids
yes, addition is commutative
thick lynx
Ah nvm
thick lynx
think in R² to simplify things
Seems true..
marsh rapids
when a plane is generated by 2 lines
Is that a direct sum?
thick lynx
when a plane is generated by 2 lines
Is that a direct sum?
Can every point on that plane be written uniquely? Yes. So that should be a direct sum, or am I tripping
marsh rapids
it is
so any 2 non parralel lines in the plane give a direct sum that generates R², right?
marsh rapids
that's not a full counter-example
thick lynx
And then pick some two distinct, non-parallel lines and represent them as subspaces
Yep that should be it
Thank you!
.close
lone heartBOT
marsh rapids
so what encouraged you to think it's true?
thick lynx
.reopen
lone heartBOT
✅
thick lynx
so what encouraged you to think it's true?
I don't have much intuition on this; considering I just read basically one definition, and then somehow it seemed as if U_1 and U_2 can only be "distinct up to a linear factor" in their span representation, so equal
marsh rapids
because you wanna consider the standard basis
can you justify that behavior using what you've read from your linalg class?
marsh rapids
I don't have much intuition on this; considering I just read basically one defin...
hence read more
you'll be much more time efficient that way, I think
rather than alternating
thick lynx
can you justify that behavior using what you've read from your linalg class?
Which behaviour?
marsh rapids
that somehow, the standard basis is to be preferred
thick lynx
that somehow, the standard basis is to be preferred
Where is it to be preferred; the problem didn't state anything about the standard basis? 
marsh rapids
but you felt that way
if you're asked to picture a U and W whose direct sum is R², you pick the standard basis right
thick lynx
Well yeah, that's how our standard coordinate system is built; with the standard basis
thick lynx
One unit along the y-axis is (0, 1), one unit along the x-axis is (1, 0)
thick lynx
why not (1, 2), (2, 1)?
Because that represents a different coordinate system than what we're most familiar with
marsh rapids
it respects the natural cartesian product nature of R²
marsh rapids
why not (1, 2), (2, 1)?
think of the dot product
poor thing would be a pain in that basis
thick lynx
think of the dot product
poor thing would be a pain in that basis
Yeah I guess we'd have to do change of basis on the vectors to our standard basis, then do the dot product, then convert back
marsh rapids
the dot product is a bilinear form, you don't need to do that
(au + bv) . (cu + dv) = acu² + bdv² + (ad+bc)u.v
why do we prefer our standard basis?
marsh rapids
Yeah I guess we'd have to do change of basis on the vectors to our standard basi...
also there's no "convert back"
thick lynx
also there's no "convert back"
Generally $\beta_A(v, w) = v^{\tr} \cdot A \cdot w$ and $A$ has the columns that represent our basis; when we're in our standard basis, then $A = I_n$ and so $\beta_A(v, w) = v^{\tr} \cdot w$, that's much easier and what we know as multiplying entry-wise
ocean sealBOT
Kepe
marsh rapids
yes
A = In is optimally simple
because the standard basis is orthonormal
that makes it extra nice
marsh rapids
\begin{problem} Let $V$ be a $K$-vector space and $U_1, U_2, W \subseteq V$ sub...
and what of this question if we ask that the spaces be orthogonal to each other?
thick lynx
Imagining it in R^2 again; you have two lines that are L-shaped
One of those lines is W and so fixed
The other line is thus uniquely defined
So it holds there
marsh rapids
and surely, the chapter on orthogonality contained statements about orthogonal complements, such as their uniqueness, right?
thick lynx
and surely, the chapter on orthogonality contained statements about orthogonal c...
The uniqueness of phi here?
thick lynx
Ah
marsh rapids
and surely, the chapter on orthogonality contained statements about orthogonal c...
no
U subspace V
U's orthogonal is {v in V | for all u in U, u.v = 0}
thick lynx
There was this
marsh rapids
it is unique
And it's a complement of U
thick lynx
Other than that, nothing else really
marsh rapids
oh beta is your dot product as a bilinear form
thick lynx
Yeah
marsh rapids
non-degenerate
so this is A being invertible
marsh rapids
The uniqueness of phi here?
what are the orthogonal abbildungs of R² with the standard dot product?
thick lynx
what are the orthogonal abbildungs of R² with the standard dot product?
The identity map
marsh rapids
anything else?
thick lynx
That one is always one, I guess
marsh rapids
yeah
thick lynx
Also phi: x -> -x
marsh rapids
anything else?
pallid scarab
(hint : you won't get them all by looking by hand. So give yourself an arbitrary endomorphism of R^2 in matrix form a,b,c,d and find conditions on coefficients for it to be orthogonal)
pallid scarab
yes any rotation of vectors of R^2 is an orthogonal endomorphism
but we're still missing one type
thick lynx
but we're still missing one type
Hmm, mirroring is still like rotating so that's not it
marsh rapids
mirroring wrt a line?
pallid scarab
Hmm, mirroring is still like rotating so that's not it
mirroring is rotating?
marsh rapids
is that really a rotation, when some vectors remain fixed?
thick lynx
is that really a rotation, when some vectors remain fixed?
Namely mirroring along a vector
So that one of them stays fixed
marsh rapids
So yeah mirroring is the last type
notice this is stable by composition
so "mirroring rotations" also count
right?
now Rafilou claimed that's it
thick lynx
notice this is stable by composition
Is "stable by composition" ~ well-defined?
marsh rapids
this is a set of functions
thick lynx
now Rafilou claimed that's it
Yeah
marsh rapids
the composition of two such functions is still in this set
thick lynx
So it's a subspace?
marsh rapids
now Rafilou claimed that's it
can you try to explain why that is?
thick lynx
So it's a subspace?
- We can scale each of these and it'll still be in this subspace, that doesn't change anything
- We can compose these and it's still in this set
- The zero-mapping is uh
marsh rapids
1) We can scale each of these and it'll still be in this subspace, that doesn't ...
prove it
marsh rapids
3) The zero-mapping is uh
uh
thick lynx
The zero mapping is not in this set?
marsh rapids
it's not
thick lynx
So it's not a subspace
thick lynx
Oh
marsh rapids
So it's not a subspace
did you do some abstract algebra?
If so, you may think about this being a group
zinc ocean
W
thick lynx
did you do some abstract algebra?
If so, you may think about this being a group
Haven't done a lot of AA but I do know what a group is
marsh rapids
then I let you check that with what we've said, we've almost proven it's a group
thick lynx
- associative w.r.t. some operation
- there is a neutral element w.r.t. to that operation
- there exists an inverse element w.r.t. that operation
For 2) that's just id
For 3) that's just the inverse of that operation; each of these operations will have an inverse because uh
We need to use $\beta(\varphi(v), \varphi(w)) = \beta(v, w)$ somehow
ocean sealBOT
Kepe
thick lynx
To show that $\varphi$ has an inverse
ocean sealBOT
Kepe
marsh rapids
that will work
thick lynx
that will work
I think we can also go the adjunct route, for $\beta(\varphi(v), \varphi(w))$ there exists a $\varphi^$ s.t. $\beta(\varphi(v), \varphi(w)) = \beta(v, \varphi^(\varphi(w)))$
marsh rapids
(also, beta is just the regular dot product here)
ocean sealBOT
Kepe
thick lynx
And this adjunct has the property that $\varphi \circ \varphi^* = \id$
ocean sealBOT
Kepe
thick lynx
So 3) is fulfilled
marsh rapids
under what conditions?
thick lynx
That beta is non-degenerate
marsh rapids
And this adjunct has the property that $\varphi \circ \varphi^* = \id$
for any phi?
thick lynx
for any endomorphism phi
marsh rapids
what's 0* ?
marsh rapids
**Kepe**
so no
among other things because it must be that phi is invertible for that to have a chance of holding
And then you basically notice that phi* = phi^-1
and that's not true in general
so no
marsh rapids
To show that $\varphi$ has an inverse
but that can be done by elementary means
thick lynx
among other things because it must be that phi is invertible for that to have a ...
Hm, but for $\beta(\varphi(v), \varphi(w))$, there always exists a $\varphi^$ with $\beta(\varphi(v), \varphi(w)) = \beta(v, \varphi^(\varphi(w))$ by this theorem
ocean sealBOT
Kepe
marsh rapids
it exists
thick lynx
Now by this theorem
marsh rapids
oh so you had things on that topic
marsh rapids
so it's a remark, it doesn't provide a proof?
thick lynx
thick lynx
So this is just exactly that
thick lynx
so it's a group
Yeah
marsh rapids
we call it On(K) (in our example O2(R))
what else did you have on it?
actually I can look myself
marsh rapids
this is page 52, you should know that
ok it doesn't seem to dive much deeper
I need to think about whether there's an accessible proof before I ask the question
let phi(v) = Av
find what the matrix A* that represents phi* is
the question was in general, not just orthogonal matrices