I keep working myself in circles on this one. I start by making the two problems or whatever
#linear-algebra
wintry steppe
hey
i dont understand how this is transformed
oooh
its (row,column,value)
alright
i don't get this part
what is IA = 0,1,3,3
why*
noble kettle
IA[0] = 0
IA[i] = IA[i − 1] + (number of nonzero elements on the (i-1)-th row in the original matrix)
source : wiki
noble kettle
🙂
pallid swallow
looks resolved
latent minnow
why no tex >.>
please tell me where im being stupid. when proving something is a vector space (over a field $\mathbb{K}$), the 5th axiom requires that scalar multiplication $\cdot$ be associative i.e. $$\forall \alpha,\beta\in\mathbb{K}\forall v\in V \alpha \cdot (\beta \cdot v) = (\alpha\cdot \beta) \cdot v$$
however, $\alpha \cdot \beta$ is not a priori defined since the scalar multiplication on the vector space is a map $\cdot : \mathbb{K}\times V \to V$. What am I missing to make the translation to the scalar multiplication on the field rigourous i.e. $$\alpha\cdot_V (\beta\cdotV v) = \alpha \cdot_\mathbb{K} \beta) \cdot_V v$$ where $\cdot_\mathbb{K}:\mathbb{K}\times\mathbb{K}\to\mathbb{K}$.
thorn robin
you said K is a field
therefor it has its own multiplication law
latent minnow
yeah thats what i wrote, $\cdot_K:\mathbb{K}\times\mathbb{K}\to\mathbb{K}$
stoic pythonBOT
Wormyjim:
thorn robin
it seems like you understand everything then
latent minnow
but in order to prove the associativity of $\cdot_V:\mathbb{K}\times V \to V$ we have to reduce this to $\cdot_K$, but only when it is acting between two scalars
stoic pythonBOT
Wormyjim:
latent minnow
which is to be honest a bit messy
thorn robin
the only thing you have to prove is (ab)v = a(bv)
there is nothing to translate this statement into
latent minnow
i dont see a way to actually prove that without defining specifically how $\cdot_V$ acts on an element $v\in V$ by the usual field multiplication componentwise
stoic pythonBOT
Wormyjim:
latent minnow
but such an argument would also require that the components themselves would also be elements of the field, which would not be true in full generality
thorn robin
I don't really understand what you're trying to say
if you know how to multiply two scalars, and you know how to multiply a scalar with a vector, then it should be a simple matter to verify that (ab)v gives the same result as a(bv)
latent minnow
but just saying "you know how to multiply" isn't rigourous
we have two different types of multiplications which are mappings from different product spaces
thorn robin
of course
I specified exactly which maps I was referring to in each case
you have one map K x V -> V and one map K x K -> K
maybe it would help to verify explicitly that a few different vector spaces verify the axiom
latent minnow
are you just saying to write $$a\cdot_V (b\cdot_V v)=(a\cdot_\mathbb{K} b)\cdot_V v$$
stoic pythonBOT
Wormyjim:
latent minnow
i feel like there is a step missing
thorn robin
that is the statement of the axiom, yup
latent minnow
i dont consider it proved after just writing it down
for concrete examples, i could show how this works
but im not sure how to go about proving it for an abstract vector space
thorn robin
what you are saying is a bit circular
you said you have a vector space and you want to prove it
latent minnow
i dont have a vector space until i've proved it
thorn robin
but in order to know that V is a vector space, you would have already had to verify it
latent minnow
im not saying i have a vector space
i have an abstract set V that i want to show is a vector space
thorn robin
let's just look at an example
can you prove that the axiom holds for the set of functions [0,1] -> R?
latent minnow
in particular i am trying to prove $\mathcal{L}(V,W)$ is a banach space, so firstly it needs to be a vector space
stoic pythonBOT
Wormyjim:
thorn robin
yes verifying that linear maps between vector spaces satisfy the axiom will be similar to the example I just gave
in either case you should be able to prove it with no trouble
let's say T : V -> W is a linear map
the definition of scalar multiplication on the space of linear maps is (cT)(v) := c T(v)
noble swallow
V and W needs to be vector spaces over the same field. Or at least the field of W should contain the field of V I suppose
latent minnow
i defined $\cdot : \mathcal{K} \times \mathcal{L}(V,W) \to \mathcal{L}(V,W)$ pointwise by, for each $f\in\mathcal{L}(V,W)$, $(\alpha \cdot f)(x) = \alpha \cdot_W f( x)$
sorry thats wrong
stoic pythonBOT
Wormyjim:
thorn robin
yup looks good
latent minnow
now to prove this is operation is associative...
thorn robin
it's one line
latent minnow
i know it should be one line
but i can't seem to write it down without using something which a priori isn't defined
noble swallow
You'll need the associativity of •_W, and the principle of identity of functions
thorn robin
((ab)f)(v) := (ab) f(v) = a(b f(v)) =: a (bf)(v) =: (a(bf))(v)
the only things I have used are that scalar multiplication is associative in W and the definition of scalar multiplication on linear maps that we agreed on
which expression is undefined?
latent minnow
even writing (ab)f in the first place
thorn robin
ab is a scalar right?
noble swallow
And since the functions coincide on each element, they are equal
latent minnow
lol im confusing myself (no shit)
noble swallow
In terms or the functional formulation:
$\cdot_V(α, \cdot_V(β,v))=\cdot_V(α\cdot_K β,v)$\
Where $α\cdot_K β$ is the element of K determined by its multiplication as a field, which is ahead assumed to be, so the argument of $\cdot_V$ is in its domain of definition.
Also for one of the distributive you'll need to take for assume an operation in K, namely the addition $+_K$
stoic pythonBOT
Mat:
latent minnow
ok i think i understand
idk why im always questioning the basics
im particularly good at doing this when it has nothing at all to do with what im supposed to be doing 😂
noble swallow
Lol, I think questioning the basics is a good thing
wintry steppe
Could someone explain how this system has infinite solutions?
This is the original system and has been reduced to the above ^
half ice
Three variables, two equations. There's infinite solutions
wintry steppe
Yes, but how? Aren't x and y equations of a line?
half ice
No, x and y are numbers here
y = mx + b is the equation of a line
For example, one solution is
(4) = (2) + 2
(-5) = -2(2) - 1
wintry steppe
I see
half ice
I got that by letting z = 2, and finding what x and y are. I could let z be whatever I want
It looks like you tried to eliminate a variable, but left the variable in
wintry steppe
I think what's throwing me off is what is said in the text: the solution set is the intersection of two non-parallel planes in space (i.e a line)
If you would be able to explain that, then that'll probably clear up my doubt
half ice
I don't know how you got the two equations. From a textbook?
half ice
,w determinant of {{2,4,6},{4,5,6},{7,8,9}}
stoic pythonBOT
Requested by Kaynex#7520
half ice
Weird way to express the solution, but yeah it's a line where you can choose an x,y by choosing a z
half ice
You can proceed to eliminate z, and I'm not sure why they didn't lol
wintry steppe
'
I already wrote the proof but I need help finding an example for when they are unequal
equality was easy because I just let S1 = S2
how about when they are unequal?
oh figured it out nvm
sleek helm
Nice
jagged saffron
I was wondering if my proof for this is accurate
Do i have to argue that the subspace cannot be infinite dimensional?
I dont think so, but I was told that I should
pale kayak
Is the perpendicular distance from (5,4,1) to the y-axis 5?
pale kayak
sqrt(35)
@humble brook
It was the correct answer
I calculated the distance using the perpendicular vector to the y-axis
Can anyone help me with this?
Why are they all wrong?
slow scroll
maybe they need the hats lol
pallid swallow
but how to type hats?
slow scroll
🤷 copy and paste from the question
pale kayak
Omg they totally need the hats ahhh
slow scroll
like forgetting the +C on indefinite integrals xd
slow scroll
it looks like you got the negation of the correct answer
-7(11, 4, -12) - (1, 2, -3) is what it was asking for
true palm
sacred vine
Hi can anyone help me understand this? the only thing im confused on is where the -7 comes from
slow scroll
-36+29 also #precalculus
gray dust
$x^2+12x+36 = (x+6)^2$
stoic pythonBOT
RokettoJanpu:
sacred vine
thanks my first time. isn't the -36 offset by the positive 36?
slow scroll
they factor the grouping
(x^2 + 12x + 36)
but you still have the -36+29 on the outside which is -7
slow scroll
np. #precalculus for this ffr 😉
sacred vine
thank you. i'll move my future questions to that channel. Have a great night!
tranquil junco
re: this question, row reduced form means that a = 1, b = 0, c = 0, and d = 1, right? even considering complex numbers i don't see how 1 + 0 + 0 + 1 can equal zero, at least without working in some wonky fields
half ice
@tranquil junco
It does not mean that b = 0
That's echelon form
Or wait does it mean that?... Hold up
tranquil junco
if b doesn't equal zero than a does
cause its reduced
so either way don't you end up at the same place?
it works for a, b, c, d = 0
but like.
thats only 1, and also trivial and dumb
slow scroll
if A isn't invertible, then you can have b = to some nonzero value im pretty sure
tranquil junco
but by definition the first nonzero term in each row has to equal 1, and all the other terms have to equal zero
otherwise it isn't reduced
half ice
Echelon form vs reduced echelon form
slow scroll
half ice
Oh. Then what do they define as reduced?
tranquil junco
all i have is this
slow scroll
so you could have a = 1, b=cnst, c=0 d=0
tranquil junco
what do you mean by cnst?
slow scroll
constant
tranquil junco
oh
slow scroll
scalar whatever
slow scroll
half ice
And that's really it, isn't it? Is there any other possible form?
tranquil junco
uh but how do you verify that a + b + c + d = 0 in this case?
half ice
That's the three
tranquil junco
isn't this not reduced?
half ice
1 -1
0 0
Is reduced. It fits all your rules
tranquil junco
i guess like i^2 and (-i)^2 for
feral mountain
idk I could only get 2 kaynex you got 3? 😮
half ice
1 0
0 1
1 -1
0 0
0 0
0 0
I like how none of them are complex
half ice
How prove those are three only three?
slow scroll
first one doesn't add to zero tho
half ice
Oh dorp
feral mountain
ye I've been trying to get a third 
tranquil junco
🔫
half ice
Maybe there is a complex one?
tranquil junco
would they consider i^2 and (-i)^2 distinct?
i wouldn't imagine so
hm
i wish i wasn't so dumb
i literally can't read
half ice
Same
tranquil junco
peak illiteracy hours
feral mountain
if a=1 then c=0
if d=1 then b=0. bad
if d=0 then b=-1. good
if b=1 then a=0=c=d. bad
:((
tranquil junco
we should just have an associative group where any a, b sum to zero
is that even a group
wintry steppe
did u get all 3
wintry steppe
identity should be 1 too
tranquil junco
1 -1
0 0
gray glen
"associative group"
slow scroll
the identity is not one tho
wintry steppe
it isnt
feral mountain
cuz that's not in rref
half ice
Oh. Is that really the answer
feral mountain
0 row needs to be under the non zero rows
wintry steppe
o shit
slow scroll
i don't think
0 0
1 -1
fits the def tho
tranquil junco
um
half ice
Yep that's the last one
feral mountain
all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix)
half ice
Lame
tranquil junco
is that in the def?
wintry steppe
o i c
feral mountain
from wikipedia
tranquil junco
that isn't in the definition provided by the book
feral mountain
oh
slow scroll
tranquil junco
wait i just fucking remembered that all groups are associativity
is something wrong with em
*Ume
*me
what the fuc
feral mountain
clearly
tranquil junco
associative group
slow scroll
nonassociative groups are my favorite groups 
slow scroll
kind of on a random note: the definition of rref in sloth's textbook is wonk because it is not unique for any matrix: rref(A) could be any permutation of the rows of the usual rref we think of. 
feral mountain
like JNF(A) isn't completely unique
crimson pond
hello, i had a math question to prove that i cant scale a point
how do i go about proving that
slow scroll
Define scale a point
crimson pond
dusky epoch
what's the definition of frame-independence?
crimson pond
Uhh it can work in any coordinate system if I'm not wrong
pallid swallow
I have an idea what we are talking about, but I'm not sure if it is the same
When we take a coordinate system, we take an origin point O
and that would change how p is defined
and the result from scaling the vector p would differ based on where O is
@crimson pond
crimson pond
I think p is a point @pallid swallow
pallid swallow
What does that mean to this sketch @crimson pond ?
dusky epoch
Uhh it can work in any coordinate system if I'm not wrong
that isn't a rigorous definition by any means
you should have had a definition given to you in class
remote spire
I'm confused, since both A and C choices give zero vector upon multiplication, which would be the true null space amongst these. Or is it both of them. (Since A is linearly dependent, I'm quite confused). The null space of C comes out to be exact (1,-1,2) but for A since it's linearly dependent it has 2 null spaces (1,-1,0) and (0,0,1)
pallid swallow
C
because they said "nullspace spanned by v"
so, the nullspace must be spanned by v
not any other vectors like (0, 0, 1)
remote spire
Alright, but I have a follow up question, while calculating the null space for c, we write the null space as (1,-1,0) but isn't 0 just anything. You could put * there and then it becomes valid for it.
I'm inclined towards c but just want to get my concepts correct.
half ice
What do you mean by "but isn't 0 just anything"?
What is *?
Null(C) = Span(v)
Null(A) != Span(v)
Null(A) contains Span(v) but isn't equal to it
remote spire
Oh alright, yeah the same ^. I was thinking if span(v) belonged to the null space of A, it could be possible answer too.
But I understand it now.
dim fulcrum
There is a circle passing through the points A (a1; a2), C (c1; c2) and
touching the axis OX. Using given-find, find the center coordinates and
the radius of this circle. Build a circle using parametric
its representation and triangle ABC, where B is the point of tangency of the circumference of the axis
OH. Also note the position of the center of the circle. Coordinates of points A and C
ask yourself.
PLEASE
<@&286206848099549185>
<@&286206848099549185>
pallid rampart
solar rock
bruh
winter reef
dim fulcrum
I will die tomorrow
plucky sapphire
You son of a gun
pallid rampart
Dog shut up
plucky sapphire
doesnt work on non helper channels
winter reef
pallid rampart
You didn’t get a ping
dim fulcrum
What a 15 min
dusky epoch
wrong channel.
raw drum
solar rock
bruh
plucky sapphire
You be nice to us, you @##@$$#@
dim fulcrum
Ok
Пожалуйста это задание нужно решить в маткаде, примерное его выполнение с другими условиями изображено на рисунке!
dusky epoch
большинство здесь по-русски не говорят
я - редкое исключение
но я, к сожалению, маткадом не владею
dim fulcrum
Окей, а в теории вероятностей вы хороши?
dusky epoch
со мной и на ты можно... в теорвере не эксперт, но в какой-то степени знакома. но это тогда уже в #probability-statistics
или в личку, чтобы не засорять англоязычный сервер русской речью
dim fulcrum
Хорошо, можно тебя в друзья добавить если у меня возникнут проблемы в этой теме?
dusky epoch
если хочешь - добавляй
dim fulcrum
Thanks
tranquil junco
why do they never read rools 
pale kayak
Could anyone help me with a vector product question?
slow scroll
heavy glacier
Not sure how dim(Im(T)) is still 2 when restricted to U, I keep getting 0
Not sure why my image of T restricted to U calculation is wrong here
cloud cedar
you did XA - XA not XA - AX
pale kayak
Can someone help me with a dot product problem?
sonic osprey
gray dust
yikes double posting, please try not to do that
pale kayak
Oh okay sorry I didn’t mean to be annoying I just thought since they are different chats the specialties might be different
gray dust
just post your q
pale kayak
I just wanted to know if I take the easiest one point (5,5,5) would the point (5,5,0) be the one we needed to find the angle between the vector and the xy plane?
gray dust
are you finding the angle between T_2 and the xy plane?
pale kayak
Well I’m trying to find the angle between any of the vectors and the xy plane but to be specific let’s do t2
gray dust
what did you try so far?
pale kayak
Kind of what I’ve already told you just using (5,5,0) and doing dot product
I don’t think that’s correct though
gray dust
dot product of what two vectors?
pale kayak
I don’t know
I just need the angle between t2 and the xy plane to do the physics here
I don’t really know how to do dot product despite taking two full college courses that covered it
I’ve spent 7 hours on 9 homework questions non stop I really just need help here
gray dust
$\langle a,b,c \rangle \cdot \langle p, q, r \rangle = ap+bq+cr$
stoic pythonBOT
RokettoJanpu:
pale kayak
Do you know if (5,5,0) is the right point to do this with though?
gray dust
that could work
pale kayak
So is the angle 50?
pale kayak
The bot keeps loading and reloading the image I don’t know what it is
gray dust
what equation did you use to calculate the angle?
stoic pythonBOT
RokettoJanpu:
gray dust
that's fine. how can you get theta from that equation?
pale kayak
Take the inverse cos-1
But I might sound dumb here but what language is that? Like how do you know what the bot will respond to?
gray dust
yes, arccos of both sides
it's latex. a kind of language that's useful for writing out math stuff. there's an online syntax reference
whatever you want the bot to show, put it in between dollar signs like i did
btw make sure your calculator is in degree or radian mode depending on which one you prefer to use
pale kayak
I have it in degree
pale kayak
I have another question but this is more physics related
gray dust
sure
pale kayak
Okay so ignore my messy handwriting but how am I supposed to get the tension on the cords if all the angles are symmetrical
gray dust
you can represent a vector as the product of its magnitude and its unit vector
pale kayak
gray dust
tbh use the symmetry of the situation. the tensions are equal in magnitude
pale kayak
I know that
I was never very strong in physics
I just don’t know how to calculate it if the angles are the same
My substitution method doesn’t work then
gray dust
remember i said the tensions are equal in magnitude
T1 = T2 = T3 = T4, that should simplify the equation
pale kayak
T1=T2=T3=T4=mg?
gray dust
no...
pale kayak
I don’t know what else to do with that
stoic pythonBOT
RokettoJanpu:
gray dust
i tell you that a=b, how can you use that to simplify the equation?
pale kayak
Substitution
pale kayak
I’m spending an hour per homework question here I really am just having a rough time I don’t understand
gray dust
$a + a = 4$ or $b + b = 4$
stoic pythonBOT
RokettoJanpu:
gray dust
do the same thing for your problem
gray dust
is it not clear that T1 = T2 = T3 = T4?
pale kayak
It’s blatantly clear
I think I mentioned that was a given the very first time I gave the problem
gray dust
then make a subsitution as i showed in the example i gave above
pale kayak
I don’t know how to find a single tension
gray dust
then make a subsitution as i showed in the example i gave above
pale kayak
Substitution to what
gray dust
$a+b = 4 \\ a = b \\ b + b = 4$
stoic pythonBOT
RokettoJanpu:
pale kayak
That doesn’t connect to my problem
I don’t understand
I’ve taken a full advanced physics course I promise I’ve done way way more advanced problems than this I’m really just having a bad time here and I can not remember how to find the tension
gray dust
then take a breather, think about my example, how it relates to your problem, the fact that the magnitudes of all of the tensions are equal, and how you can use that to make a substitution EXACTLY in the same way that i showed above
pale kayak
Can you explain yourself in trigonometry at least I don’t understand what you are trying to explain with the symmetry of your a+b problem I know basic algebra
That’s not my problem here though
I just don’t understand how to find the tension at all so I don’t know what I am substituting
If you look at a separate problem of mine where the angles are different
gray dust
$a+b+c = 6 \\ a = b = c \\ a + a + a = 6$
stoic pythonBOT
RokettoJanpu:
pale kayak
I. Know. How. Substitution. Works. Please.
gray dust
then i have no idea how you can't make a substitution for your problem
pale kayak
I’m struggling with the physics here
If I try to do the same with two strings that have the same angle then it’ll cancel out
gray dust
why are you showing me a different problem?
pale kayak
Because I wanted to show you that I only understand how to find tension when the angles are different
Please help me when the angles are the same
I know T1=T2=T3=T4
But I have no fucking idea how to substitute that into
gray dust
@pale kayak what if you had to make a ton of substitutions like this?
$a+b+c+d+e+f = 6 \\ a=b=c=d=e=f$
stoic pythonBOT
RokettoJanpu:
gray dust
no idea AT ALL?
pale kayak
@gray dust
I really appreciate you helping me
Is this what you meant?
@gray dust
That answer came out wrong
gray dust
@pale kayak what’s the value of mg?
pale kayak
2254
gray dust
What did you use for g
pale kayak
9.8
gray dust
My concern would be you’re not using enough sig figs in your angle
pale kayak
I used the full 35.26438968
I redid it and got 976.01 but that’s still wrong
I’m sorry you’ve been helping me with this for 2 hours @gray dust
My teacher just emailed me and told me that finding an angle isn’t even an option so all this work has been for nothing
vital egret
Hello am I free to post my question here
frosty vapor
yes
native ore
Ok
so I just came out of summer vaca
and I cant math
help pls solve for a and b
gray dust
you know how to add vectors?
gray dust
yeah that's right
stone drum
Yes
pale kayak
@gray dust I hate to kick a dead horse and I know I’m stupid but I still haven’t gotten the answer for that question
Can someone help me with this problem?
Please
I’ve found the angle to be ~35
Tz is 2254
The unit vector direction is <1,1,1> or 1/sqrt(3)
The length from (0,0,0) to (5,5,5) is sqrt(50)
Please I really want to understand this problem it’s due in an hour and I don’t want to just use the answer key
native ore
how about you use the answer key and then understand it
so ur not stressed about the time limit
pale kayak
That would lose me 8 points on my homework
I only get the answer key if I forfeit all the points
I don’t want to give up please
stone drum
@native ore Dunno. Could be crappy software. I see nothing obviously wrong with your answers.
native ore
my linear algebra prof must of made a msitake then?
pale kayak
Can anyone help me with my problem?
mint surge
Write the second column of LB as a linear combination of the columns of L
Here is L and LB
My confusion stems from what to do with these numbers. I dont quite understand what a linear combination is I guess?
I know the second column of LB is -50 44 '
Would it just be -50*7 + 44*9 as one answer, then -50*-8 + 44*9 for the second etc?
woven oasis
@mint surge a linear combination is the sum of vectors times their scalars is equal to one vector for example : b = s1v1 + s2v2 + s3v3 + ... + spvp
blissful vault
what's the difference between basic solution
and general solution
and a linear combination
woven oasis
well a general solution they want you to solve for x1, x2, x3, ect... right? or for how many variables u have
blissful vault
i have 3 variables
2 equations
but like, i'm talking about the format
cuz i got the answer
1 0 1.8
0 1 3/5
0 0 0
(it's a coefficient matrix)
woven oasis
so the general form would be solving for x1, and x2
and the linear combination would be
would be finding a sum of vectors
that would add to one new vector
blissful vault
huh
the last row is another variable row
not the answer row
i can' do iii
but the other 2 confuses me
woven oasis
so you would have s [1;-1;-1] + t [2;3;8] = [3;0;3]
the sum of those two vectors equals that one vector as a linear combination
s = 1.8, and t = .6
when you typed rref (a) into your calculator
blissful vault
?
woven oasis
u got it down to reduced echelon form right?
blissful vault
i don't have a calculator
also the problem says that it's a coefficient matrix
not an augmented
matrix
so there's no answer
we shouldn't be able to get s = xyz
woven oasis
oh right
blissful vault
homogenous means everything = 0
so
i get x1 + 1.8 x3 = 0
x2 + 3/5x3 = 0
thus
x3 = s
x2 = -3/5s
x1 = -1.8s
but which solution is that?
general?
i'm just confused about terminology'
blissful vault
... wolfram
blissful vault
then basic form.. also "basis"
idk what they mean
they should be different though
woven oasis
so
a basic solution i think would be something like this
wait nvm
i gotta go ask someones else for help, if nobody else responds ill try
but i think a basic solution is
nvm ill ask my teacher, i'm learning this stuff these are good questions
just to double check
blissful vault
ayt
woven oasis
I know that we have the general solution form right, but im not sure what they mean by basic so yeah im gonna ask
slow scroll
the basic solutions are the two solutions the algorithm gives you. The basic solutions should span the space of all of the solutions
or at least I think. I haven't heard the term "basic" solution before and (ii) is kind of redundant because the basic solutions you write down in (i) are the basis for the space of solutions
woven oasis
yeah, thats what im thinking too, most of the terms in linear algebra mean the same thing /:
makes things super confusing
slow scroll
column space = range = image is my favorite lol
blissful vault
thanks, cya
wintry steppe
Question: can a vector be a linear combination of other vectors if one of the coefficients is a free variable?
sonic osprey
What
hollow spear
how do I find these C1,C2,..Cm?
Now, Gauss Jordan method or Echelon won't work?
hardy blaze
if a 2x3 matrix is all 0's
all the entries are 0's
is that rref
i dont think it matters its 2x3
a mxn matrix with all 0's is rref right 🤔
yes it does
hollow spear
What if there are 10^9 values in B matrix?
hardy blaze
yes
hollow spear
The calculation would be tedious?
Is there any other way to check if the sequence can form the constant K?
hardy blaze
wait were u asking my question or the other guys question ..
idk wat the other qeustion is D: 😔
hollow spear
n is a linear combination of x and y if and only if n is a multiple of the gcd(x,y)
undone garnet
$A,B \in M_n, AX=B$
if $\det A = 0$, can we imply that we cannot have only one solution for X?
stoic pythonBOT
Nguyễn Thành Trung:
slow scroll
well, inverses are unique
undone garnet
so if det(A) != 0, X = A^(-1)B is only solution
otherwise, det(A) = 0, we cannot have only solution?
late glen
Can someone explain what is happening in this? Im having trouble understanding why T is in the inner product.
cloud cedar
Tv is a vector in W, so you can take the inner product of Tv and w
you define T* such that you can apply it to w to get a vector in V where you get the same inner product
slow scroll
@undone garnet thats right. if det(A) = 0 it will either not have a solution or have more than one solution
late glen
thx
glass imp
hey guys i just joined this chat today
i really need help with this problem and i’ve done just about everything to find it and i still can’t figure it out
if any of you guys could help me it would be really appreciated if not it’s okay
half ice
Before anything, note this isn't linear algebra. The channels aren't so busy though, so we can do this here
glass imp
oh i apologize
half ice
They give you two points on this line, they are:
n = 4000, p = 32
n = 5000, p = 28
Can you compute the slope of the line?
half ice
Now, when I say "y", I mean "line height". In this case, p represents how high the points are
32 - 28 is the rise! That's the difference in height of the points
glass imp
so we have 4
half ice
You want the rise, and you want the run as well
half ice
So slope is?
glass imp
4/-1000?
half ice
Exactly!
Now, they want that with three decimal places, but I like that you wrote that as a fraction
half ice
You wrote it correctly above
Rise divided by run
4 / -1000
You'll need a calculator to get the decimal approximation of that
glass imp
-0.004?
glass imp
is that the final answer ?
half ice
That's the slope. One might say we're halfway there
That's the m in p = mn + b
Now we need b. Know how to get that?
glass imp
not exactly haha i haven’t done this type of work in a long time
and now i actually have to learn it because i’m in college
half ice
You have
p = -0.004n + b
Where p and n are related by this equation, but b is some unknown that we still need to find
glass imp
how do we find b
half ice
You want to put one of your points into the equation. That is, sub n = 4000, p = 32 in
Or, you can sub the other point in if you like
They both should give the same b value
glass imp
32=-0.004(4000)?
half ice
32 = -0.004(4000) + b
half ice
Since b is the only thing you don't know in the equation, you can solve for it
glass imp
16?
half ice
32 = -16 + b
32 + 16 = b
48 = b
half ice
If you'd like to, give it a try
half ice
You're interested in the slope, and y-intercept. Both of them we've found
m = -0.004
b = 48
Put them back into your equation
p = -0.004n + 48
With that, you can know how to set the price of the shirts, if you want to sell n of them
half ice
Glad, hope it helps with some other ones
In summary, they gave us two points, we drew a line between them.
glass imp
yeah that’s what some of the questions ask me to do
scarlet hamlet
hey could i share my solution for an assignment to verify if i did it properly
pallid swallow
(yeah, people have made that mistake in abstract algebra before)
Sure, show your work
(and the question)
scarlet hamlet
itll be awks if people here have the same assignment too 👀
pallid swallow
just show it lol
noble swallow
If that concerns you, use ||spoiler syntax|| 
scarlet hamlet
pallid swallow
|| here's how you would type something to spoiler ||
scarlet hamlet
like for a) can i just say -- consider zero matrix and be on my way?
@pallid swallow sounds good, thanks
scarlet hamlet
likethat was my initial solution, then i thought hey that can't be an assignment question if it takes like 2 mins to do
so i played around with it
i think my a) is corrent but b) is a little wonky
wintry steppe
linear algebra is hella baller
me n the gang be solvin equations with it when we hit the town
pallid swallow
well if you are good you can do assignment questions in 2 minutes
scarlet hamlet
yes but i really am not good
wintry steppe
^
scarlet hamlet
oh did i remove the question
pallid swallow
yeah you did
scarlet hamlet
pallid swallow
and I'm having a hard time reading your solution
scarlet hamlet
LOL sorry uh ill write bigger
pallid swallow
I see you make it (1, 0, 2) and (1, 1, 1)
noble swallow
And if you can turn on the light in your room lol
scarlet hamlet
ok facts but im also lowkey rushing to class rn
ill take a better picture
sOon
but we could tlak about maybe how you would approach the question?
wintry steppe
good luck 🅱lessed student
noble swallow
Oh nevermind it was my phone with low battery
scarlet hamlet
tbh im really suspect about another question because
it seems way too straightforward
thats literally how i know im doing something wrong
if i can do it on the first try
but anyways, if anyone can help, just ping me and ill pop right over (if im not in class)
would rly appreciate ❤
or should i be using like the question channels
noble swallow
Actually you are only required to prove or disprove the existence of such matrices, so finding them is unecessary effort
If they exist
pallid swallow
(b) is straightforward, rank nullity theorem gives dimension of the domain is at least 4, but we only have vectors in R3
pallid swallow
ah, no wonder
scarlet hamlet
mhm -- they're making sure we understand the basics first 👀
noble swallow
What you call dimension theorem is rank-nullity theorem
scarlet hamlet
oh ok LOL
great ill understand something in class finally
@pallid swallow wait why 4?
noble swallow
Someone calls it rank+nullity theorem
pallid swallow
heh
scarlet hamlet
im not seeing where the dim of domain is at least 4
pallid swallow
well, the domain is R3
hmm, wait, the dimension of the domain needs to be exactly 4
but the nullspace says the domain is of dimension 3
because we are working with R3 vectors
noble swallow
In a) A could be some m×3 matrix, but b) tacitly assumes that A is a 3×3 matrix when aks for col(A) to be V.
dim(col(A))+dim(null(A))=n=3
sonic osprey
What have you tried?
blissful vault
well i think it only has unique solution when at the end the rank and the number of variables is the same
but i'm not sure how to prove it
sonic osprey
What happens in the other situations?
In this case, can the rank be equal to the number of variables?
blissful vault
if there is not enough rank there is infinite solutions
if there is too much rank it doesn't have sol'n
sonic osprey
For part (a), it only asks if the solution COULD have a unique solution or no solution
So all you need to come up with are examples
of systems with 4 equations and 6 variables that either have a unique solution or no solution
Or say that one of these can't happen
blissful vault
hmmm is there a proof for that?
last assignment i gave examples but teacher wrote
example != proof
sonic osprey
Because it depends on what the question is asking for
part (a) is not asking for any sort of proof
It is asking whether or not this situation COULD occur
Well, I think the point is that
in this case, an example is a proof of the statement
Because it's asking whether or not there exists such a system
When you're trying to prove something for all systems, then just showing it for an example doesn't work
blissful vault
okay i'll just write random matrices that work then?
noble swallow
Though there's no example with a unique solution, so supporting the answer in this case would require a proof
blissful vault
huh
noble swallow
What you said about rank and number of variables is true, it can be justified with rank nullity theorem
Adding that if the system has a unique solution, then dim(ker(A))=0
blissful vault
alright thanks
how about no solution?
it's not possible since it's a homogenous right
noble swallow
Right
Had the question been for a non-homogeneous, an example could have been found with no solutions
blissful vault
i'll just write it's impossible since the max rank is 4
and show that in a matrix like the one in the problem max rank is 4
noble swallow
It seems ok to me
blissful vault
alright thanks!
leaden ermine
Hello, could anyone help guide me with a very simple proof? I am trying to write a proof for "Let u,v be two vectors of R^2. Show that either they are linearly dependent or that they span the whole of R^2. Where would I start with such a proof and what kind of proof would be best suited for this example? Right now it feels as if im just writing down definitions of linear dependence vs linear independence for 2 vectors which must then span R^2
half ice
@leaden ermine
Basically, show that one NOT happening implies the other
leaden ermine
Hmm, so an if not Y then X
half ice
IF linearly independent, THEN span all of R^2
IF there's a point they don't span, THEN they're linearly dependent
Those statements are contraposatives though. Proving one proves the other. So, choose whichever one you want
leaden ermine
Ok awesome thanks man! Im going to try to write it out now 🙂
So if we use the contrapositive, we are basically taking P -> Q . Where P represents linear independence and Q represents spanning all of R^2, and turning it into "If not Q" -> "Not P", which is saying if u,v does not span all of R2 then it is not linearly independent?
leaden ermine
In order for a linearly independent set of vectors to span a vector space R^n, where n=2, it must contain at most n vectors. Otherwise, to potentially span a vector space R^2, a family of vectors of V must be linearly dependent and contain at least n+1 vectors
Since n=2 our set is either linearly independent & spans R^2, or linearly dependent and does not span R^2.
slow scroll
The first paragraph there sounds wonk.
leaden ermine
Hahaha yeah, maybe im thinking of the contrapositive wrong. Also, It was suggested for this proof I simply show if X is true then Y cant be true, and if Y is true X cant be true
blissful vault
when counting rank
do we also count the last row?
aka
100|3
010|4
001|5
000|1
is this rank 3 or rank 4
slow scroll
is that an augmented matrix?
pallid swallow
depends if you are finding the rank of the augmented matrix or the rank of the matrix that represent the coefficients of the variables
slow scroll
@blissful vault its not clear whether you are talking about the rank of the matrix behind the bars or including the stuff to the right of the bars, but I typically like to count columns to find rank. idk if that would be easier for you
pallid swallow
It should be specified when asked to find the rank
like "Find the rank of the augmented matrix", which means, include the last column
Solve for Ax=b, "Find the rank of A", which if you make an augmented matrix, it does not include the last column
blissful vault
- yes it is augmented
- So if asked for rank of augmented matrix, i count the 000|1 row too? since 000|1 = impossibble
blissful vault
but rank doesn't depend on that
slow scroll
the augmented matrix is rank 4 tho
blissful vault
ok thanks
slow scroll
you could say that if rank(A) < rank(A | b) then A is inconsistent @blissful vault
gilded crest
Hey, can anyone recommend me an online linear algebra lecture series that is proof oriented? I'm taking a proof based linear algebra course rn, but the professor doesn't lecture like at all and I've been doing p bad. I've seen the 3b1b series, but it obv doesn't really help much with proofs as much as geometric intuition.
sonic osprey
Well what's the usual way to show that a long list of statements are equivalent?
paper egret
show one thing is true
sonic osprey
?
sonic osprey
You should think about what statements being equivalent is
paper egret
am I supposed to prove that the identity matrix is all of the above? a b c d?
sonic osprey
Not really
paper egret
yea ok im sry, im not understanding what it's asking in the first place
sonic osprey
You should think about what statements being equivalent is
paper egret
i'm not sure what that means
only thing i know is that it's an n*n
which means the number of rows = number of columns
statements being equivalent means they hold the same logical equivalence, but written differently
oh wait this is reduced row echelon form
hol up
sonic osprey
The usual way to think about it is that two statements are equivalent if each implies the other
paper egret
the only thing i can think about is it being a RREF and a n*n matrix, which implies that all the entries column or row wise, are 1's
but i'm not sure how to formalize it
sonic osprey
Anyways, the usual way to show that a long list of statements are equivalent
is to show that (a) implies (b)
then (b) implies (c)
then (c) implies (d) and finally that (d) implies (a)
This shows that any single one of the statements implies all the others, so you can get away with proving only four statement
sonic osprey
Most of the time people will list them in the order which it's easiest to prove alphabetically
paper egret
ok
wait is it true, that RREF and Echelon form is somehow related to linear transformations
cuz I can see a relationship, but I'm not sure how to describe it
but at the same time, something feels off
wintry steppe
this is less of a linear algebra problem and more of understanding how to pack pieces on a chessboard
cobalt tartan
Hrm
How would one find a matrix with a given nullspace?
Like in general, not in a particular example
Like say that we're given some set of vectors V = span{v_1, v_2... v_n}
And we say that this spans some nullspace of a matrix A
How would one find the matrix A?
Like the nullspace is the set of x such that Ax = 0
Then if we want a matrix with a given nullspace what we're saying is that all possible combinations of V are in x and vice versa righT?
thorn robin
extend v_1,...,v_n to a basis
set Av_j = 0, let A do anything else to the rest of the basis you found
it's a projection that collapses V
cobalt tartan
What does extend v_1... v_n to a basis mean?
thorn robin
ok I mean assuming the v_1,...,v_n you gave me are linearly independent
thorn robin
then you can find w_{n+1},...,w_N so that v_1,...,v_n,w_{n+1},...,w_N form a basis of R^N
cobalt tartan
Oh so what you're saying is that V = span{v_1, v_2... v_n} only spans R^n, and we want it to be able to span R^N?
thorn robin
well it wasn't clear how big of a matrix you wanted at the start
usually I'd use n for the dimension of the whole space.. but you used it up so I had to pick another letter lol
cobalt tartan
Ah
Well, I mean generically
However generic you can make it 😂
Like I'm just curious about how the whole thing works
Because I had a question about it in my homework and I didn't really understand the idea behind it but I could crunch the numbers to do it lol
thorn robin
but yes, to write down a matrix you need to know each of the columns
requiring that Av_j = 0 for j = 1,...,n tells you that the first n columns should be zero, in the right basis
then just do whatever you want with the other columns
cobalt tartan
Ah
So if you have some mxn matrix, and you have n vectors that form your nullspace then your matrix is pretty much already restricted to be Av_i = 0 for i = 1..n?
thorn robin
to find the corresponding matrix in the basis (1,0,...,0), (0,1,0,...,0),...,(0,...,0,1) you will need to use a change of basis formula like B = PAP^{-1}
cobalt tartan
But if we have a mxk matrix, k > n, then we just need Av_i = 0 for i = 1...n and then for all i > n, your thing can be anything?
thorn robin
since I have used a very specific basis in order to write down such a simple matrix that sends all the v_j to zero
well if you are saying that A is mxn and your nullspace is n dimensional... that's the whole thing right?
that's why I used a bigger N than n, otherwise you are sending everything to zero
cobalt tartan
Oooh
thorn robin
But if we have a mxk matrix, k > n
yeah that's right
same thing I was saying in reply to your previous sentence
cobalt tartan
So if you have a mxn matrix and your nullspace is n dimenisonal... then everything is in the nullspace right?
thorn robin
right
cobalt tartan
So then the vectors v_1... v_n form the first n columns of your matrix?
And then the rest can be anythinG?
thorn robin
yes, if you use v_1,...,v_n as a basis
thorn robin
nope
you will get a different projection depending on how you extend your v_j to a basis of the whole space
cobalt tartan
Oh so they don't even have to be linearly independent?
thorn robin
yes I assumed they form a basis
thorn robin
I mean you need to create a basis of the whole space
cobalt tartan
What do you mean?
thorn robin
to write down a matrix you need to know what it does to every vector
hence you need to know what it does on a whole basis
so v1,...,vn,w{n+1}...,wN should form a basis of R^N
writing down a matrix that looks like the identity, but the first n ones replaced by zeros, will give you a projection of R^N onto span{w_j} which sends span{v_j} to zero
make sense?
each column of a matrix corresponds to what it does to one vector of the basis
cobalt tartan
Oh
Hrm
A matrix of size mxk can be represented as A = [a_1 a_2... a_n... a_k], with each a_i being a column vector of size m right?
thorn robin
yeah
cobalt tartan
Then a nullspace for it is the set of Ax = 0, which is really that each a_i * x = 0 right?
thorn robin
what is a_i * x? that doesn't look right
cobalt tartan
It's the ith column vector of A dotted with x
thorn robin
but a_i has size m and x must have size k
to find (Ax)_j you must use the jth row of A, not the jth column
cobalt tartan
hrmm
I'm not quite sure that I follow the why, but my initial question, the how, seems to be satisfied
cobalt tartan
Hrm
thorn robin
the main idea is just: use the v_j as part of a basis, write down a very simple matrix that sends these v_j to zero in this basis, then use change of basis formula B = PAP^{-1} to find matrix B in standard basis (1,0,...,0),....,(0,...,0,1) that sends these v_j to zero
cobalt tartan
Is there some way that you can like... put this into an example with a matrix? :x
I'm sure that I'll understand it if I see it like that form but words don't seem to be cutting it :x
thorn robin
sure
maybe say v = (1, 1)
make this into a basis of R^2 by adding w = (0, 1)
in the basis {v, w} the matrix (0, 0; 0, 1) sends v to zero and w to itself
because in the basis {v, w} v has coordinates (1,0) and w has (0,1)
now we want to write this in the standard basis
so we need to find the change of basis matrix between {(1,0),(0,1)} and {(1,1),(0,1)}
it should have v and w as columns so it's like (1, 0; 1, 1)
you can find that the inverse is (1, 0; -1, 1)
thorn robin
you need to multiply those 3 matrices together, you should get (0, 0; -1, 1)
which you can see sends (1; 1) to zero
np, seeya 😋
cobalt tartan
Hrm
Given a matrix B and a matrix A, if the columns of B are linear combinations of the columns of A, does this mean that dimCol(B) <= dimCol(A)
I think it does because the columss of B would be a subset of the columns of A right?
slow scroll
right, the column space of B would be a subspace of the column space of A. I think another way to think about it is that AC = B for some matrix C. rank-nullity says that rank(A) + dim(ker(A)) = columns of A.
The null space of B must contain the null space of A and C. If the null space of B contains vectors that are not in the null space of A, then the dim(ker(B)) > dim(ker(A)). Therefore the rank (dimension of column space) of B must be less than that of A.
noble swallow
@wintry steppe may we do it in the context of a vector space? Or would you set it in an affine space?
half ice
The system you have above is two equations:
x - 6z = 5
y + 3z = -4
What's y in terms of z?
Oh we're colliding
noble swallow
@topaz plover please use a free channel, a question is still in progress here
noble swallow
Alright pascal, then for instance we might formalize the question letting AB=v, AD=w
And AC=u
wintry steppe
This is just to simplify it a bit？
noble swallow
Then DC=u-w
Yes, to rewrite the problem in a compatible notation with a vector space
BC=u-v
wintry steppe
Then can I just use the axioms in the bottom right corner?
noble swallow
Yes sure
You can try to rewrite the statements in this notation we adopted to see things more clearly
wintry steppe
Okay, I will play around with that, I’ve gotten to the post where I am questioning everything
noble swallow
It's a positive attitude
Alright, I hope you can come out with something
I have to go now unluckly, sorry for the short assistance
topaz plover
im not sure how I would
write this out
in a matrix
I've come up with
x + y +z = 5600 and 0.08x + 0.096y + 0.144z = 606.40
but I think it needs one more equation including the last part of the problem
hey
slow scroll
@topaz plover
topaz plover
s
slow scroll
z = 600 + y. Thats another one
slow scroll
$\begin{pmatrix}
1 & 1 & 1 & 5600 \
.08 & .096 & .144 & 606.4 \
0 & -1 & 1 & 600
\end{pmatrix}$
stoic pythonBOT
kxrider:
slow scroll
does that make sense>