since noone has posted for a while
#linear-algebra
slow scroll
Its an isomorphism if you can come up with another linear transformation that sends the outputs of T back to (x,y)
boreal crescent
isomorphism implies linear and invertible right
i get that, i was trying to use the other condition they defined
that its an isomorphism if ker(T) = 0 and im(T) = W
slow scroll
sure. if you wanted to, you could find the matrices for those linear transformation and compute the kernel and column spaces (images) of each.
boreal crescent
ok so the last one is not because the matrix isnt invertible
second last one is an iso
idk about C
B is
idk about A
slow scroll
hint on A and C: are they linear transformations? i.e. does T(x + y) = T(x) + T(y) for all vectors x and y and T(ax) = aT(x) for all scalars a and vectors x.
boreal crescent
wait
so A wont be cus its kernel is the empty set right
T([x,y]) will never be [0,0]
slow scroll
so A wont be cus its kernel is the empty set right
In an abuse of definition way, yes. But if T were a linear transformation, it would always at least contain the 0 vector. Since T isn't a linear transformation, the notion of kernel doesn't exactly apply
boreal crescent
T has to be LT for it to be iso though right
so to answer this specific question wouldn't the reasoning work
slow scroll
yea. Its has to be a LT to be iso. It definitely fails that.
slow scroll
nor multiplication. Yea B and D look good
boreal crescent
awesome, it works. my course is 'Inquisitive based learning', which implies that we have to do the reading and we spam questions in class.
so currently im trying to learn this with a few other dudes and we are confused as fuck lol
slow scroll
or well by closed under addition/multiplication, I assume you mean it fails linearity T(x+y) = T(x) + T(y) and T(ax) = aT(x).
boreal crescent
yea
slow scroll
oh yea those flipped classrooms. Haven't had one of those yet :p
boreal crescent
its rewarding when you learn it on your own, but its so much fucking work lol
ive become somewhat of a regular on this channel at this point
dusky epoch
what problem
brittle orchid
I don't understand the very basics of this topic, i.e. what do you mean by:
"R is a reflexive if for all x (which belong to X), xRx"
what exactly does xRx refer to
To be specific, I get lost in the introduction
Line #3
brittle orchid
Also, what does it mean for something to be "even" in those notes
quasi vale
Divisible by 2
brittle orchid
Thanks
idk I feel so lost in this topic...
I feel like I can't understand the basic definitions, for instance, I have no idea what this means:
Let R be the relation on Z defined by aRb if a+b is even e.g. 3RS
Also, what does if y=x is a multiple of 3 mean?
Specifically the y=x bit
quasi vale
@brittle orchid What it means is that a,b are some elements and are related by the relation a+b, which has a name R.
And it means that a,b are integers.
So a+b also results in an integer.
brittle orchid
So when you say they're related to each other by the relation a+b, what do you really mean?
That there is another set of elements (a+b)?
and that set is called R?
quasi vale
One sec, lemme edit.
They're not related to each other per se, a has nothing to do with b. It's just that we create a relation between them, in this case a+b.
quasi vale
Yeah.
brittle orchid
Gotcha
Also the other thing which confused me, which, as it turns out I might have copied down wrong lol
In my notes it says define R on Z by xRy if y=x is a multiple of 3
Turns out it was meant to be y-x
Just to confirm, the former wouldn't make sense, right?
quasi vale
Yeah
brittle orchid
Thanks
quasi vale
Although I'm confused here, xRy if y=x is a multiple of 3. Idk tbh
brittle orchid
Apparently it was meant to be if y-x is a multiple of 3
Which seems easier to understand x)
Like, if it was y=x is a multiple of 3, idk what that actually means
quasi vale
Oh
brittle orchid
does that mean y = x = 3z
quasi vale
no that doesn't make sense
brittle orchid
I'm assuming I just copied down wrong
quasi vale
y=x is a multiple of 3 doesn't make any sense, but y-x is a multiple of 3 does make sense
What it means is x,y are some integers.
When you subtract these integers, you get another integer which is a factor of 3 or divisible by 3.
brittle orchid
Yeah that makes sense
The other thing I wanted to ask is regarding the following example (pic incoming)
in part 3 when trying to prove transivity
it says: (near the bottom of the picture)
Adding z-x=3(w+u) where w+u is an integer
Should I assume the results from part 2 have been used to prove that y-z=3u => z-y=3U (for a different U)
quasi vale
You need to use the previous results.
brittle orchid
I just meant that should I assume that one of the two equations has been changed using the previous result, in this example
y-x=3w (i)
y-z=3u (ii)
for some integers w,u
Since it later says adding z-x
quasi vale
No equation has been changed.
brittle orchid
Then it's not using those two equations?
quasi vale
In part(a) and (b) we just use two elements namely x,y. in part(c) we need a third element, so we name it z and start all over again.
Yes it's using those 2 equations
brittle orchid
since (i)+(ii) wouldn't give you z-x
brittle orchid
Aren't we using the result that z-y=3(-u)
I mean if you subtract eqn (ii) from (i) you get:
z-x=3w-3u
brittle orchid
Yeah
quasi vale
so w-u = an integer
brittle orchid
I understand that
quasi vale
3(w-u) also an integer
brittle orchid
and that makes perfect sense
but if you read the second last line
it says
adding z-x=3(w+u)
not w-u
quasi vale
?
brittle orchid
So my question is, should I assume that what he (the professor) meant to write is "z-y=3u"
because that would make sense for some other u, wouldn't it?
as shown from 2)
quasi vale
Transitive property goes on like this (a,b), (b,c) so (a,c). So it should've been y-x, then x-z, then y-z
brittle orchid
so I have to show xRy and yRz as part of showing the transitive property?
brittle orchid
yeah
quasi vale
Depends on the relation
brittle orchid
how does the implication symbol fit into this though
xRy and yRz => xRz
how is that "implied" if you get my question?
brittle orchid
isn't that just showing 3 separate results?
quasi vale
What it's saying is if you have xRy AND yRz(both together), then you always have xRz
brittle orchid
ahh I see
quasi vale
It's using two results actually and from those we get the third.
brittle orchid
Ahh
So you have to use those two results only to show the third
Thanks a lot, btw
@quasi vale
quasi vale
np
chrome current
I've been stuck on this one for a good hour now, feels like I'm missing something really obvious. Please help.
half ice
@chrome current
Let's say X is [a,b,c]. Multiplying everything out, we get the system:
-5a + 6b + 2c = -35
1a + 9b - 7c = -23
4a - 1b + 3c = 38
We can write this back into a block matrix:
[-5 6 2 | -35]
[1 9 -7 | -23]
[2 -7 3 | 38]
half ice
Np. So the beauty is that we can solve that with the "normal" techniques you'd have learned so far. Row reduction comes to mind
What ended up happening was that I transposed everything, and got a form we are more familiar with
chrome current
But when I do row reduction, what result do I want to end up with?
half ice
I suggest googling "row reduction to solve system of equations" as this needs a lot of pictures to explain lol
Row reduction is a common technique you'll want to know it
chrome current
Will do! Thanks for the help
odd kite
@rotund jetty I think I solved your trace problem from yesterday if you still don't have it
the method I was hinting at should work
rotund jetty
@odd kite I solved it. Ty tho!
obsidian jackal
hello i need some help
i am confused here
why does the equation equals to the determinant of the matrix
the equation is different if i were to go by cofactor expansion along the first row
sweet lily
no it's not
if you go by cofactor expansion along the first row you get exactly that
no, wait, by bad
the mistake is between the first and the second step
it should be -y(a_1 - a_2)
sweet lily
you multiply both sides by -1
sweet lily
@obsidian jackal np. And disregard the "it should be -y(a_1 - a_2)". The original is correct
rotund jetty
N(T) is the null space/kernel, and R(T) is the range
sweet lily
hint: let V,W be vector spaces. V and W are isomorphic iff dimV=dimW
rotund jetty
yeah - but I don't have that the dimension of V1 is the same as the dimension of N(T)
sweet lily
N(T) depends on the function T. That's what you'll use to start constructing T
sweet lily
np 👍 keep at it
restive hound
if a linear system in R space has 1 free variable then it has infinite solutions right
slow scroll
yes. for example a system of one equation in two variables
ax + by = c
has an infinite number of solutions (they form the line y = (-ax + c)/b )
agile gyro
if it has a solution
gleaming topaz
Given that a vector has the coordinates [2;-3] in B', how can it be written as a linear combination in B?
I got it to [16/5;-7/5] but apperantely that's wrong
why?
quartz compass
what did you do to work it out?
gleaming topaz
Honestly it was on a test so I don't have the calculations rn but I tried controlling by getting that the vector 2v1-3v2=[-1;5] right and then 16/5 * [1;2]-7/5 * [3;1]=[-1;5]
so shouldn't that be right or am I completely wrong
@quartz compass
quartz compass
that seems right to me, that's how I interpret the coordinates to mean
$2v_1-3v_2$
stoic pythonBOT
Merosity:
quartz compass
that's the vector represented in the B' basis
gleaming topaz
Yeah
quartz compass
then the inverse of B times that will get you the coordinates in that basis which is what you got
stoic pythonBOT
Merosity:
quartz compass
idk you don't have the "right" answer or anything, just that it's wrong eh?
gleaming topaz
The question is literally find the coordinates for [v]B for the vector v=2v1-3v2
quartz compass
well as long as you can explain your reasoning to them for why you think you have it right, you should be good
gleaming topaz
Can I ask you one thing in PM?
quartz compass
I usually don't like to talk in pms, especially if it's about math
nimble egret
if it helps
imagine an extra column of 0
see if you can derive some form of relationship between the vectors
gleaming topaz
How is this the wrong answer for defining linear independence? I answered this where omm is the same thing as iff https://gyazo.com/70b19fd71b6d0768987abcc5eb72f079
gleaming topaz
fuck me thanks, but when writing omm which is equal to if and only if
does it not imply only?
dusky epoch
the iff is in the wrong place
you should have written
[equation] iff all lambdas = 0
gleaming topaz
Lol, thanks
So what I've written is that, that equation always has the trivial solution but needed to specify that it can only have the trivial solution?
gray dust
in order for $\brc{v_1,\dots,v_n}$ to be lin indep, yes the above eqn you wrote must only have the trivial soln $\lambda_1=\dots=\lambda_n=0$
stoic pythonBOT
RokettoJanpu:
dusky epoch
key word: ONLY
gleaming topaz
I got it, thanks once again
wise furnace
I have a question with change of basis (no matter how hard I try I seem to not be able to compute anything!): We are asked to choose a basis B and compute [T]_B, where T is a linear transformation. The specific question is attached, and the solution is also attached. My question: how do we get the 4x4 matrix from the given basis? (I can't convince myself if I'm wrong or right- there's no understanding here)
vast torrent
Okay I see what they did @wise furnace it's not obvious
Would you know how to do this problem with a function on column vectors?
wise furnace
I think so, I seem to get those correct 50% of the time^ (i'll explain):
wise furnace
We choose a basis, then for each basis element we apply the transformation and see where it maps too, and those are the new columns of matrix
vast torrent
T(b¹),T(b²),...,T(b^n)
Yes exactly
The same idea here, the columns are MA for each A in the basis but
wise furnace
Problem is, when I do T(b^1), and so on using the matrix multiplication MA, I get 2x2 matrices that I don't know how to represent together as a 4x4 matrix
vast torrent
Then they took each 2x2 MA and picked an isomorphism to send each to a 4-tuple
However the usual isomorphism for sending nxn matrices to n²-tuples is to stack the columns on top of each other
Here they stacked the rows, ie the columns of the transpose
For some strange reason
wise furnace
Hmm, let me work out T(b^1) and see how this stacking is chosen
also- why does stacking (in such a way) result in an isomorphism?
vast torrent
I think that's what they chose, im trying to do it in my head
Good question. I'll leave that as an exercise for the reader
wise furnace
So I stacked my T(b^i) and when we took our 2x2 matrices and 'read them' top-down from column 1, then then concatenate to get the 4x4 matrix
vast torrent
That's the way it's usually done. I think they did it by rows and then transposed them, but im not good at doing these multiplications in my head
You do it for me
wise furnace
I think actually it's an isomorphism because usually bases are represented as column vectors, so as long as some ordering is chosen (and we stay consistent) on the 2x2 matrices to represent them as column vectors, it will preserve algebraic structure and hence show isomorphism?
wise furnace
Assuming T is LT, and yeah the T(b^i) are the 2x2
vast torrent
This operation is called vec() sometimes
wise furnace
I'm guessing something on the lines of "vectorising the matrix"? (meant by vec())
vast torrent
Oh wait
They did use vec()
Columnwise
Mental arithmetic isn't my strong suit
Yes exactly
wise furnace
That's why you do maths 😉 (I'm the same- I use python's numpy though to do all the numbers)
Mental arithmetic isn't my strong suit
Thanks dude, makes much more sense now 🙂
vast torrent
I let computers do my arithmetic for me
Anyway this is in a sense the best isomorphism because it satisfies
tr(A^t B)=vec(A)^T vec(B), for A and B square same dimension
So it gives an inner product on matrices that works the same as the dot product on vectors
wise furnace
I just checked this statement, not sure it's correct as then determinant of the LT are not the same if we chose to use rows instead of columns
I think actually it's an isomorphism because usually bases are represented as column vectors, so as long as some ordering is chosen (and we stay consistent) on the 2x2 matrices to represent them as column vectors, it will preserve algebraic structure and hence show isomorphism?
vast torrent
Actually this might work the same for other isomorphisms
Isomorphisms don't have to preserve determinants
You're thinking of isometries
Anyway I'm not sure how many of these properties hold for a row vectorization map
wise furnace
I unfortunetaly haven't gone over isomorphisms much (they were just told to us an algebraic structure + bijection), why does
So it gives an inner product on matrices that works the same as the dot product o
tr(A^t B)=vec(A)^T vec(B), for A and B square same dimension imply isomorphism?
vast torrent
But we usually think of vectors as columns so I'm not really interested in this row vec
It doesn't, I'm just saying why we like vec
One reason
wise furnace
I was just interested with the idea of mapping matrix bases as column vectors and operations we can use for that^
Oh right- sure, it has special properties useful down the line
wise furnace
That sounds very next level- i'll try not to worry much about it until then, then
bronze tangle
Not sure if I should put this on help but I'm struggling with this question for a while now and it's supposed to be easy
Given two sqaure matricesA and B, prove or provide a counter-exmaple: If A and B have the same characteristic polynomial, then they have the same rank.
wise furnace
(\begin{pmatrix} -1 & -1 \ 1 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}) how about this?
stoic pythonBOT
dinobear32:
vast torrent
those doesnt have the same char poly
wise furnace
x^2?
dry spear
how come when trying to find the distance of a particle, with function l(t), to the distance of a hyperplane, you sometimes use only the velocity vector, other times you don't
like for one question I had to find the distance when the particle was in the middle of 2 planes, and I had to use the position vector and velocity vector
but I had another question where I had to find the distance between a particle parallel to a plane, and I only needed to use the velocity vector
solar osprey
So there is this problem where
They ask to show that the only subspaces of R is R itself and {0}
I gave it some thought
It is very intuitive
But I dont know how to translate this intuition into actual math
gray dust
do you have the defn of subspace?
solar osprey
Yea
A subset S of a vector space V is a subspace of V:
- if the zero vector O is an element of both S and V
- for any scalar K, and any vector X in S, K•S is in S
- for any two vectors u, v in S, u+v is also in S
gray dust
do you have a good feel for what these things say? see if you can apply them to your q
solar osprey
I dont have a problem proving that {0} and R are subspaces of R
That is a trivial matter
But I am not sure about proving that they’re THE ONLY subspaces
sonic osprey
Take some subspace
Well subspace has to contain 0 by the definition
but consider if it contains anything other than 0
What happens in that case
solar osprey
If we add two elements of this subspace
We are not guaranteed to still be in that subspace
Thats the intuition
half ice
So let's say we have a subspace of R that contains 0 because it has to, but also contains something other than 0
sonic osprey
Further let's call this element x
gray dust
sonic osprey
To be a subspace, what other things must this subspace contain
gray dust
let $S\ne\brc{0}$ be a subspace of $\bR$\let $a\in S$ with $a\ne0$\let $b\in\bR$\try to show $b$'s presence in $S$\eventually show $S=\bR$
stoic pythonBOT
RokettoJanpu:
gray dust
glhf
wintry steppe
Interesting, we covered up subspaces just today on our linear algebra class
lime scaffold
Hi !
I can't find my mistake...
I take A and B, polynomial
C = A*B
I'm trying to find B such that C = A * B knowing A and C
A can be writed like a_0 + a_1 * x + a_2 * x^2 + ...
B : b_0 + b_1 * x + b_2 * x^2 + ...
C : c_0 + c_1 * x + c_2 * x^2 + ...
Following the formula c_k = sum for m from 0 to k {a_m * b_(k-m)} :
c_0 = a_0 * b_0
c_1 = a_0 * b_1 + a_1 * b_0
c_2 = a_0 * b_2 + a_1 * b_1 + a_2 * b_0
c_3 = a_0 * b_3 + a_1 * b_2 + a_2 * b_1 + a_3 * b_0
...
More esier to read :
c_0 = a_0 * b_0
c_1 = a_0 * b_1 + a_1 * b_0
c_2 = a_0 * b_2 + a_1 * b_1 + a_2 * b_0
c_3 = a_0 * b_3 + a_1 * b_2 + a_2 * b_1 + a_3 * b_0
...
It's seem to be equivalent to solve :
* =
[a_0, 0, 0, ...] | b_0 | | c_0 |
[a_1, a_0, 0, ...] | b_1 | | c_1 |
[a_2, a_1, a_0, 0, ] | b_2 | | c_2 |
...
But this doesn't work..
Y can resolve the system, there is as much equations than unknown variables but with the B found, C != A * B ...
dusky epoch
@real plaza you should ask your prof what you'll be tested on if you're so inclined
email?
@gray dust they might mean problems like "here's a thing we constructed, prove it's a vector space"
gray dust
i had a feeling that's what was intended to be said, i just had a gut reaction to that particular phrase
dusky epoch
when is the test
bronze tangle
x^2?
@wise furnace Thanks dude, I was looking at both of these yesterday but made a calculation error
solar osprey
@gray dust Okay so we assumed that S != {0}, thus there exists an element b in S such that b != 0. Since we assumed S to be a subspace, then by definition of a subspace, for every K in R, we have that K•b is in S. And since b is any element != 0, 1/b exists. Now consider any element r in R and take K = r•(1/b). Therefore, K•b = r. And we just said that K•b is in S for any scalar K in R. It follows that R is a subset of S since for every element r in R, r is in S. Moreover, it’s pretty clear that S is a subset of R (I mean I don’t even see why one should attempt to rigorously prove that). Since R is a subset of S and S is a subset of R, then S = R.
sonic osprey
yes
solar osprey
Feels good 😂
Not satisfying tho, I personally hate solving with hints
but yeah sometimes, you just need a little push
gray dust
looks nice 
Moreover, it’s pretty clear that S is a subset of R (I mean I don’t even see why one should attempt to rigorously prove that)
at the start we let S be a subspace of R, just from that S is a subset of R
storm carbon
I'm having trouble with the wording of this question
could someone tell me what its trying to say
where s is a scalar.
(a) Write out the matrix form of this transformation
tender island
hello I need help with quaternions/rot-matrix stuff
So I have 2 coordinate systems. I have 3d rotation for first one in form of quat/matrix/euler etc. and I need to get it's rotation matrix in second system.
Second system is gotten by swapping Y and Z axis from the first one. How do I get the rotation matrix in second system?
wintry steppe
are the basic variables always going to be the slack variables?
gloomy arrow
If you have a 3x3 matrix and wanted to find how they span R^3, how would you do that??
wintry steppe
hi
im having trouble showing that the basic solutions to a system of equations are linearly independent
can anyone give me a hint ?
nimble egret
🤔
wintry steppe
do you understand my question?
wintry steppe
oh i forgot to mention that they are homogenious
i guess my question can be reworded as
im trying to prove that a basis for the null space is the basic solutions
i showed that they span
so snow i have to show they are linearly independent
nimble egret
so what's your working so far
wintry steppe
idk im kind of stuck
i was trying by contradiction at first but im sure if i can get one
so i have c_1x_1 + c_2x_2 + ... + c_nx_n = 0 and i said one of them is non zero
wlog x_1 = -1/c_1 (c_2x_2 + ... + c_nx_n)
so i wrote one basic solution in terms of the other ones
in not sure if i can get anywhere with that or not
@nimble egret
gray dust
@gloomy arrow show the matrix row reduces to the identity
wintry steppe
I don't understand what they mean with 'Ax=b has a solution iff b is a linear combination of the columns of a.'. Could someone explain this to me?
dusky epoch
a linear combination
what is it that you don't understand here?
the statement itself? or why it's true?
wintry steppe
The statement itself
So x only has a solution if b can be retraced from a combination of Ax combinations?
vast torrent
$\mathbf{Ax} = x_1 \mathbf a_1 + \ldots + x_n \mathbf a_n$ where $x_i$ are the components of $\mathbf x$ and $\mathbf a_i$ are the columns of $\mathbf A$
stoic pythonBOT
gfauxpas:
wintry steppe
Ah yea I understand now
But why is that a highlight in my book, is that not the entire point the entire time?
vast torrent
if and only if
the only if is less obvious
or maybe the if is less obvious
idk, but they not be both obvious to the reader
wintry steppe
hmm okay
vast torrent
also, important iff statements generally get their own presentation in math books
"refer to theorem 2.33"
wintry steppe
I suppose its like a conclusion in this case
Because it comes after the explanation
Thank you for the explanation by the way 🙂
keen pond
Hi guys. Does anybody know some software to solve a system of equations (matrix-like), but when the coefficient are letters. I need to solve a 6×6 matrix, but the program I did and the ones I know only works with numbers, not with letters. Thanks in advance
spiral sonnet
Which ones are in row echelon form and which ones are in reduced row echelon form? In my eyes there are none that are in reduced row echelon form
quasi vale
@keen pond There could be online websites maybe?
dusky epoch
@spiral sonnet do you know the definition of reduced row echelon form
spiral sonnet
uhhh yeah there's the staircase and there has to be zeros above the non zero entry and the non zero entry has to be one
I think F might be in reduced row echelon form
spiral sonnet
I mean reduced haha
There's two one's in the first row
This is what I think of when I think of reduced echelon form
wintry steppe
I think values are allowed if there is no pivot in the column
spiral sonnet
can you only have pivot columns in reduced row echelon form?
wintry steppe
No you can have columns without a pivot, those are 'free parameters'. Someone correct me if I'm wrong, I'm currently taking Linear Algebra for the first time as well 🙂
dusky epoch
This is what I think of when I think of reduced echelon form
nope
you should learn the actual definition of rref
wintry steppe
Ann am I correct?
dusky epoch
you CAN have non-pivot cols of course
spiral sonnet
I thikn I phrased my question wrong. Can you have pivot columns in echelon form
Also is E neither? I'm thinking that the third row is what would make it neither
wintry steppe
I suppose I meant not a pivot but a leading entry.
So it's possible a column does not contain a leading entry which means it can contain values. Because only if there is a leading entry in the column there have to be zero's above and below it.
wintry steppe
@keen pond sympy if you don't mind writing a couple lines of python
keen pond
@wintry steppe I already did so using Mathematica, but thank you
dusky epoch
@lilac kindle no, this approach is not correct
it will fail for any matrix that isn't 2 by 2
which a lot of matrices aren't
lilac kindle
@dusky epoch i dont know how to approach this question then
could you give me a little hint ?
dusky epoch
can you tell me what "a is an eigenvalue of A with eigenvector v" means?
lilac kindle
it means det(A - λI) = 0 and eigenvector means Av = λv
dusky epoch
what's lambda?
lilac kindle
it's the eigenvalue
dusky epoch
okay so you either didn't read the statement i asked you to explain the meaning of, or ignored the part where at no point did i mention any variable named lambda.
no, i was expecting you to say "Av = av and v ≠ 0"
lilac kindle
okay, i get that
lilac kindle
i still dont know what to do
dusky epoch
well now your problem asks you something about A^-1
so what's the most natural way to bring that into the equation we've got
lilac kindle
but i dont know what A^-1 is
dusky epoch
you don't need to.
you don't know what A is either.
now maybe deal with that little a that feels like it's on the wrong side.
maybe there's something else we could multiply by on both sides.
lilac kindle
so now i have v/a = A^-1 v since a is a number i can do that
but i cannot divide vectors
dusky epoch
you aren't dividing by a vector
division by a nonzero scalar is perfectly fine - division by a is simply multiplication by 1/a, nothing else.
$A^{-1}v = \frac{1}{a} v$.
stoic pythonBOT
Ann:
lilac kindle
yep, but i still have v/a = A^-1 v and i need to get A^-1 by itself
dusky epoch
no you don't.
lilac kindle
im confused
dusky epoch
you aren't asked for A^-1 itself.
just an eigenpair for it.
and if you look closely at the equation i just wrote
you should see that you already have it
dusky epoch
not in this notation, but yes. it's in the form of <matrix><vector> = <scalar><same vector>
lilac kindle
thank you!
radiant meteor
spiral sonnet
If an augmented matrix has no free variables does that mean there's only one solution?
half ice
It typically doesn't. The question likely tells you that A and B are special
wintry steppe
i was just reading this book, and it said that the characteristic of a field is the amount of times you need to add 1 to get 0
how is a characteristic greater then 0 even possible
sonic osprey
What's the characteristic of a 12 hour clock
wintry steppe
would a 12 hour clock be considered a field though?
sonic osprey
no but it doesn't really matter
characteristic is a concept that holds for more things than just fields
wintry steppe
well in that case it would be 12
sonic osprey
If you want an example of a field, you can consider a 5 hour clock
This is actually a field
Under the operations of addition and multiplication (mod 5)
and like you noted, its characteristic would be 5
and 5 > 0
wintry steppe
ohhhhh
alright thank you
also would a "negative number" just be 5 minus the number
in that scenario
sonic osprey
When we think of things with positive characteristic
i.e. non-zero characteristic
We usually don't think of some type of "ordering"
because you would want things like
if a > b, then a + 1 > b + 1 to be true
but you can't really have that since like maybe you would want 4 > 3 but then adding 1 would give you that 0 > 4 which
And there are proofs that there's no nice way to put an ordering
wintry steppe
ok
but according to the definition of a negative number, it is a number such that x+y=0 where y is negative of x
so you said 4+1 = 0
therefore 1 is the negative of 4?
sonic osprey
Oh
There's a difference between being a negative number and the negative of a number
usually we call the latter thing the additive inverse of a number
But yeah, 4 = -1
sonic osprey
try to prove this from the field axioms
wintry steppe
well since 0 is a unique number such that x+0 = x
wait
also x+(-x)=0
and if x=-x
x+x=0
and -x+-x =0
x+x+x=0
x(1+1+1)=0
so if the characteristic was 3
then this would be possible without x = 0
sonic osprey
I'm not sure how you got x + x + x = 0 from the thing
sonic osprey
sure
wintry steppe
since x=x (i think) , 1+1+1=1
sonic osprey
I mean, yeah sure that works
sonic osprey
Yes
wintry steppe
that would be like binary
sonic osprey
kind of
spiral sonnet
if I want to solve an augmented matrix does it have to be in reduced echelon form?
smoky lagoon
which one
smoky lagoon
does the red mean its wrong
gray dust
it may help to recall $\log_b(a)=\frac{\log(a)}{\log(b)}$
stoic pythonBOT
RokettoJanpu:
smoky lagoon
so those answers are not correct
digital garnet
yea
smoky lagoon
is it a coefficent matrix or an augmented matrix?
@gray dust i need to get a constant to show the relationship and i dont think $\log_b(a) is a constant
digital garnet
whats the difference again?
smoky lagoon
coefficient matrix is just the coefficients of the system of linear equations]
augmented matrix is the coefficients and the right sides
so in t hat case, you have a coefficent matrix
each coefficient corresponds to a column in the matrix
digital garnet
leo ur answer is 1 i think btw
gray dust
what constant would make log5t = log7t
try writing an eqn that's more mathy than this
digital garnet
isnt it one?
smoky lagoon
yerr
smoky lagoon
to find the rank just go to row echelon form
and fidn the number of pivots
@gray dust im confused
digital garnet
leo t = 1
smoky lagoon
i'm not solving for it i need to show that they're linearly dependent
I need to find two constants that i can apply to the two different functions (in this case they're being used as vectors) to make their sum equal to zero
smoky lagoon
that requires actual thinking and i dont want to do that right now
digital garnet
o
gray dust
tell me what it means for two things to be lin dep
smoky lagoon
c1v + c2w = 0 where c1 and c2 are nonzero constants
gray dust
rewrite this so you only got 1 const to worry about
smoky lagoon
(c1/-c2)=v/w
gray dust
rewrite using just 1 const
smoky lagoon
c1/-c2 is just another constant
gray dust
call it c
smoky lagoon
c=v/w
so i just need to show that the relationship between v and w is a constant
their ratio
gray dust
v=? w=?
smoky lagoon
v = log5(t), u = log7(t)
gray dust
run w/ that
smoky lagoon
it goes to natural log 7 over natrual log 5
but i dont know why
i forget that rule
gray dust
base change rule. always useful
smoky lagoon
fuck
i should remember that
can you prove that 3 equations are linearly independent using the same logic?
spiral sonnet
is there a rule of thumb when trying to get a matrix in reduced row echelon form?
gray dust
you can but it's tedious and make sure you're not dividing by what may be 0
smoky lagoon
because I can show that
= a constant
but whats the other way? coujld i show by transitivity that they're linearly depdendent
where each of those is a function I need to verify
I guess I would also know that 2^2t = (2^t)^2
wait no
that’s wrong
slow scroll
@smoky lagoon wut is ur question?
gloomy arrow
this would be r4 right?
gloomy arrow
I have no idea for this one
sometimse right
Because you could have all pivot columns
Or you could get 0= something on the bottom
slow scroll
yep, so its sometimes.
Hypothetically, what if b=0? Does Ax = b always have a solution?
gloomy arrow
That would be the next one lol
slow scroll
wasn't asking u 
but yea, x=0 iis always a solution
gloomy arrow
So always not sometimes for this one
thin bloom
I'm having hard time resolving 0 = 0
gloomy arrow
Ax=0
thin bloom
How do I solve 0=0?
gloomy arrow
That would always have a solution ebcause you can always reduce it down and you would get something like 0=0 on the bottom right
thin bloom
Hmm, maybe it is tautology
gloomy arrow
So ax=0 is always consistent?
slow scroll
Ax = 0 is consistent iff you can find a solution. What is a solution to Ax = 0?
smoky lagoon
@slow scroll i'm able to show that 2^t is linearly dependent to 2^t+2.7, but i cant shw that it is for 2^2t or that 2^t+2.7 is linearly dependent to 2^2t
thin bloom
0=0, then 0-0=0-0
slow scroll
@thin bloom stop
gloomy arrow
0
thin bloom
Sorry
slow scroll
yes, so its consistent
thin bloom
@smoky lagoon consider definition of linear dependence
slow scroll
x=0 is always a solution.
Not sure what you mean? Sure, you can reduce the matrix down. x = 0 would still be a solution
gloomy arrow
But its asking if Ax=0 is always consistent
smoky lagoon
C1V+C2U+C3W=0
slow scroll
@smoky lagoon thats all you have to do. 2^{t +2.7} = 2^{2.7} 2^t.
thin bloom
Yep, and what are the VUW in this condition
slow scroll
sure, there could be other solutions
but since x = 0 is always a solution, its never inconsistent
thin bloom
What can't you show?
slow scroll
you don't need to
thin bloom
That 2^2t and 2^t is independent?
smoky lagoon
they're dependent
gloomy arrow
But in an absolute sense, could you have an A or an x that would make ax=0 inconsistent?
slow scroll
no you can't
smoky lagoon
gloomy arrow
Having infinite solutions is still considered consistent right?
slow scroll
yes
gloomy arrow
Okay
smoky lagoon
show that they are linearly dependent
gloomy arrow
So then Ax=0 would always be consistent
thin bloom
Is there a and b for a(2^2t) + b(2^t) = 0 @smoky lagoon
Ah, you need to revise your definition of linearly dependent
slow scroll
yep always. It has to do with A being a linear transformation.
A(x) = A(x) implies A(x) - A(x) = A(x-x) = A(0) = 0
thin bloom
Could you reiterate def. of Linear dependence?
smoky lagoon
there exists a non trivial constant c1 and a non trivial constant c2 such that C1*V +C2 * U = 0 where V and U are vectors
slow scroll
take a simple example
{(1, 0), (0, 1), (0, 2)}
is linearly dependent because (0,2) = 2*(1,0). That's all you have to do.
thin bloom
What about 3 vectors?
smoky lagoon
i mean right now ive been trying to prove it with transitivity
slow scroll
ur way overthinking my dood
thin bloom
Definition of linear dependence in 3 vectors? @smoky lagoon
smoky lagoon
as far as i know it would be the same thing but just
C1V+C2U+C3W=0 where c1, c2, c3 are non-arbitrary constants and V,U,W are vectors
thin bloom
But it's always true when c1=c2=c3=0
smoky lagoon
that's the trivial solution
I need to prove
that they are linerally dependent
I know that the trivial solution is valid
thin bloom
So what solution do you need for linear dependence?
smoky lagoon
any non trivial solution
smoky lagoon
C1V+C2U=0
smoky lagoon
All I know is that V and U are linearly dependent
if i can show that either V and W are linearly dependent, or U and W are, then it follows that they all are
thin bloom
@smoky lagoon?? Why
smoky lagoon
very simple answer that i did not see
thin bloom
XD, always happens
I guess you saw that! Every great advancement happens like this
Don't punch yourself, this happens every time with everyone
smoky lagoon
smh myead
yeah its not due until wednesday even
but im trying to get all my work done today so i can spend tomorrow reviewing calc 3
sigh
i got a 100 on the first quiz which is just half of an old exam from last year but still i do not feel ready at all
gloomy arrow
Last question I need help with. I dont know the approach to this one
gloomy arrow
I got 3,0,0
stoic pythonBOT
Requested by Kaynex#7520
royal star
damn, we have some magic bots in here
half ice
Giving two equations:
x1 - 3x3 = 9
x2 = 3
half ice
We should let x3 be a free variable. x1 = 3x3 + 9
gloomy arrow
Right
half ice
But then x3 is free and you can set it. So the last number is 1
half ice
x1 = 3x3 + 9
x2 = 0x3 + 3
x3 = 1x3 + 0
gloomy arrow
yeah I knew that and I still put 0 smh
So then a1 would be 3, a2 would be 0, a3 would be 1
half ice
Oop. Sorry I didn't arrive sooner
half ice
Ya
half ice
In order to multiply AB, A needs to have a width equal to B's height
In your example, A has a width 3. So, x is from R³.
dark harbor
Yeah that will give you the answer
bright otter
but now I’m stuck at the back substitution part
gloomy arrow
@half ice OH I see
dark harbor
NO SOLUTION
bright otter
I tried No solution
dark harbor
Because it should be an identity matrix
half ice
I rarely do matrix multiplication without picturing the above thing. It's super helpful
dark harbor
Hmmm..
bright otter
Says it was wrong
dark harbor
You could do it by hand😄
bright otter
Yea I tried by hand and got up to
x + 2z = 0
y - z = 0
w = 0
And I have to write the terms in a
half ice
,w row reduce {{1,1,1,1},{2,3,1,-2},{3,5,1,0}}
stoic pythonBOT
Requested by Kaynex#7520
dark harbor
Yeah but isn’t it suppose to be a identity matrix?
bright otter
,w row reduce {{1,1,1,1,0},{2,3,1,-2,0},{3,5,1,0,0}}
stoic pythonBOT
Requested by Brother E#2684
half ice
But you do get a few equations:
x = -2z
y = z
z = z
w = 0
So there's infinite solutions and you can generate one by plugging in a value of z
bright otter
Where’d you get z=z?
Oh never mind
But now how would I write this is terms of a?
Is it
2a
a
a
.... and would I leave out w?
half ice
Oh, they just let the free variable by a
But you do get a few equations:
x = -2a
y = a
z = a
w = 0
Don't leave out w, you want to know what it is. It just happens to always be 0
half ice
Np. Feel free to ask if there's anything else
bright otter
Definitely will be back.
hidden sable
True or false: suppose ||a|| = 3, ||a +b|| = 4, and ||a −b|| = 6. Is it possible to find ||b||? Explain.
nimble egret
🤔
nimble egret
what can we write |a + b| as
nimble egret
which is?
hidden sable
how do you get to dot products with only the result avaliable
nimble egret
what's a dot a?
hidden sable
yes
nimble egret
so |a + b|^2 = ?
hidden sable
a^2 + b^2 + 2ab?
nimble egret
you should be using . for your products
and magnitude symbols
but yes
|a|^2 + |b|^2 + 2a . b
elder robin
If I have a gradient vector of a function, how can I use that the estimate a point on the original function
for example f(0.9999, 1.0002, -1.9999)
gray dust
set up a linearization of f around whatever point you took grad(f)
elder robin
Second q: If I find the gradient vector of a function and want to find the equation of the line tangent to a point, (a,b,c) I just do (a,b,c) + t(gradient_vector)
I haven't heard of linearization
I'll google though
The gradient vector is the direction tangent to a point right?
nimble egret
@hidden sable i'm doing |a + b|^2
hidden sable
oh ty
nimble egret
do the same thing with |a - b|^2
hidden sable
i did, but im not sure if it is correct
nimble egret
what did you get
hidden sable
|a|^2 + |b|^2 - 2a.b
gray dust
grad(f) encodes f's maximal rate of change as well as the direction along which that maximal rate of change occurs
nimble egret
actually that's fine
hidden sable
would i then do a systems of equation?
hidden sable
thats what i got
nimble egret
you'll probably quickly realise a problem
if you attempt to solve this system of equation
hidden sable
is there another way to do it?
nimble egret
the problem is not really a problem
it'll give you insight into whether this is possible or not
hidden sable
doesnt b = -5/3
nimble egret
why?
hidden sable
is vector a not 3?
nimble egret
the magnitude of vector a is 3
hidden sable
(|a|^2 + |b|^2 + 2a.b) - (|a|^2 + |b|^2 - 2a.b)
nimble egret
ok
nimble egret
ok
hidden sable
hidden sable
well then... im stuck
nimble egret
you need to distinguish between a^2 (which doesn't really make much sense) and |a|^2
hidden sable
whats the difference?
nimble egret
well
a^2 doesn't really mean anything
without further context
and |a|^2 is the magnitude of a
and then squared
hidden sable
im confused
doesnt the systems of equation make (|a|^2 + |b|^2)s cancel out?
you wouldnt need the |a|^2 right?
hidden sable
is the issue at the 2a.b part?
hidden sable
since it is vector multiplication
hidden sable
but not yet?
nimble egret
what's the final equation you have
hidden sable
[3].b = [-5]
nimble egret
🤔
hidden sable
since a = 3?
nimble egret
there's a geometric formula for dot products
thin bloom
Maybe you should be aware, vectors are not numbers
nimble egret
^
once you write the formula in terms of the geometric formula
you'll notice the problem
thin bloom
Multiplication of vectors is nontrivial
hidden sable
im not quite sure what the formual is
thin bloom
Vector multiply vector is only scalar
hidden sable
formula*
thin bloom
You don't need to think of that formula now
So how would you define vector multiplication @hidden sable
thin bloom
Linear combination?
hidden sable
thats what my professor gave us
or dot product of the rows
whether or not you can solve for b
|b|
mhm
thats what the helper did
but im trying to find |b|
if its possible or not
hidden sable
no
i also got sqrt17
that doesnt work either
unless im doing |a+b| wrong
no
the professor assigned this without going over this
hidden sable
since you cant do regular multiplication, i dont know what to do instead
i guess i dont know how to do dot product
but a is 3
so its [3 0 0]?
nimble egret
??
river jasper
hidden sable
its my 4th class
hidden sable
can you give a quick explanation?
hidden sable
is the a inside the sqrt vectors?
nimble egret
yes
well
ok
it's a vector because i said it is
if i was writing this down
they would use different notation
hidden sable
ok
nimble egret
and you really need to write what type of multiplication you are doing
when working with vectors
nimble egret
yes
you should be putting a dot when you are doing your multiplications here
or the other notation as shown above
boreal crescent
i know C is not closed under addition
and D is not injective
A and B seem to be linear and bijective, so where am i going wrong?
ok nvm got it. A is not iso since Dim(V) does not equal Dim(W)
wintry steppe
Can anyone give me an example of non complanar vectors in R^3 for example?
thorn robin
i j k