linear-algebra

[fringe cave]

it's nothing to learn really, it's just making it so that you have something like x = whatever

then ax+by = whatever

and so on

so it's easy to look at without requiring knowing that k

[modest jasper]

I don't know how to do that though lmaoo 😂

[fringe cave]

lmao

I'd recommend doing the adding and subtracting with the top two to get rid of z

try that

[modest jasper]

Hmmm, okay.

[fringe cave]

so you just have x and y, which is much more managable

[modest jasper]

Alright.

Thanks 😄

[warm sluice]

So a cross product simplifies to ABsinθ
And a dot product simplifies to ABcosθ?

[native lodge]

those are different representations of it, ABsinθ only gives magnitude, but the cross product actually gives a vector that is normal to the two input vectors

ABcosθ is a scalar, just like the dot product

[warm sluice]

The three-hand rule only applies to the Cross Product however right?

[native lodge]

if we had three hands, that would be interesting, but let's call that but its conventional name...the right hand rule

[warm sluice]

Oops lol I meant that

I see

Thanks!

[native lodge]

you can find the direction of the vector produced by the cross product using the right hand rule...at least up to a certain point. If your vectors are at arbitrary orientations, it can get difficult

[warm sluice]

So I'm assuming you can't do that with dot products?

[native lodge]

dot products are scalars, no directions to worry about there

[warm sluice]

Ok goodie

[astral junco]

what does a non-negative matrix mean?

can't seem to find what it means on google

i mean the negative of a matrix just has all elements signs switched right

so i can't figure it out what this means

[dusky epoch]

context?

[astral junco]

says "non-negative diagonal matrix" on this slide

[dusky epoch]

hhh

do they give any examples of SVDs later on

[astral junco]

i'll check

[dusky epoch]

bc like... maybe they mean like

all entries nonnegative?

[astral junco]

gotcha yeah that might be it. i'll see if i can find an example

[wintry steppe]

it would have to mean that all entries of Sigma are nonnegative (Sigma has nonzero entries only on its diagonal; some entries on the diagonal may be 0)

[unborn wedge]

Heyy can someone help me with smth

[brittle juniper]

[unborn wedge]

I have a true or false exercice and im trying to verify

In R^4 there exists two vectorial spaces that intersect at 0

Its true right?

Cuz r4 is a vectorial space and it includes 0

?

[broken hawk]

do you have the precise wording of the question?

because this seems too easy

[brittle juniper]

you may send the thing in its original language

[broken hawk]

for any vector space V that isn't only zero, all of V, and {0} are two subspaces intersecting in 0

@unborn wedge

[unborn wedge]

Il existe deux plan vectorielles F et G dans R4 tel que F inter G = 0

It is easy im in first year haha

[brittle juniper]

En dimension finie, on a toujours existence de supplémentaires

[unborn wedge]

@brittle juniper also "certaines applications linéaires R^2 ---> R^3 ne sont pas injectives est vrai aissi no?

[broken hawk]

vrai

f(v) = 0

[unborn wedge]

Thought so

[brittle juniper]

haha

[broken hawk]

hey I've had enough years of french to read french math :P

[brittle juniper]

nice

[unborn wedge]

I second guess myself a lot dont laugh at me xD

[broken hawk]

don't think that laighter was directed at you

[solemn lantern]

I damn near need an instructor for my current assignment... Probably be about 30 minutes to an hour tops if someone is willing to help me. Going through some pretty mentally unstabling things in my life right now, but I also have homework and my major to focus on still. Any help is greatly appreciated, and if I had money to pay for your services I would, but that's college life for you.

I'm actively following off a PDF for my assignment, and I'm using google sheets as my digital paper as it is an online class. Both will be linked accordingly.

[split parrot]

if I take sum of elements of a row/column multiplied by cofactor of some other row/column why is the result 0

a11C21 + a12C22 + a13C23 = 0
can anyone explain to me why?

[half ice]

@solemn lantern
Feel free to post any questions or pictures here

[solemn lantern]

Here is the PDF for the assignment I'm working on;
https://tinyurl.com/MAT85-A3

Here is what I've done so far...
https://docs.google.com/spreadsheets/d/1Gj3H4Aw7z1KrTMUS_foBUtmuY14SqADFQ31iQanfG1A/edit?usp=sharing

I just need to know if what I've done so far is correct, and then I'll continue on it for a bit until I have another question.

[half ice]

Can't seem to access the spreadsheet. Are you unsure about any of the questions?

[solemn lantern]

Try this one maybe

https://docs.google.com/spreadsheets/d/1Gj3H4Aw7z1KrTMUS_foBUtmuY14SqADFQ31iQanfG1A/edit?usp=sharing

[half ice]

Nvm got in

[solemn lantern]

And I'm unsure of my formulas and general understanding if a formula is linear or not.

I feel like I'm correct, but I just like to know for certain.

Once I'm sure I have the first few parts correct, then I'll move on to the slope and y intercept part of the assignment, etc.

[half ice]

@solemn lantern
Acme obeys 31500(1.05)ⁿ¯¹
Oceanic obeys 33000 + 1500(n - 1)

Oceanic is linear, acme is not

[solemn lantern]

Test?

Odd

My comment was immediately deleted lol

So I do have that correct alteast. It kinda made sense anyways just due to the fact that it's going up by a percentage, which is going to become steeper and steeper of a climb in salary as the salary increases. Whereas for Oceanic, it's a constant rate of 1500 per year.

I'd assume making something of similarity to the shape of a J over time.

ƒ(x)=m(1.05)^n-1 and ƒ(x)=m(1)+1500(n-1) then?

m representing the base or new base amount

Wait, why would it be n-1? If n is a representation of years, Wouldn't it simply be, for instance, ƒ(x)=31,500.00(1.05)^1 (for 1 year), which would equal etc?

[half ice]

,calc 31500(1.05)^1

[stoic python]

Result:

33075

[half ice]

@solemn lantern
As you can see, that gives you year 2

[solemn lantern]

Not based on the amounts I have. That correlates with the first year for Acme Corp for me. The second year I calculated 33,075*1.05 = 34,728.75

Because year 0 is when he starts, then using ^1 calculates for year 1, etc

[half ice]

Oh I see, my mistake

[modest jasper]

Does anyone know how I can answer this question?

[half ice]

Yeah, then what you have should work

[solemn lantern]

No worries haha

[half ice]

@modest jasper
What methods can you use? Can you compute a matrix determinant?

[modest jasper]

We can use any method we want.

In terms of what I know how to do, I don't know how to compute the determinant, but I can compute the row echelon form @half ice

[half ice]

Okay, then do the row echelon form. You should be able to reach the identity on the left, but keep track of if you ever divide by (k - a)

If you are forced to divide by zero, you can't get rref

[modest jasper]

The identity on the left?

I'm sorry, but I'm confused...

[half ice]

The identity matrix
[1 0 0]
[0 1 0]
[0 0 1]

As you do when getting rref

[modest jasper]

What's rref?

[half ice]

Row reduced echelon form

[modest jasper]

Oh, okay.

But how can I do that if k is on the diagonal?

I can't change k to 1, can I?

[half ice]

You can! You'll have to divide by k

It won't be just k though. Start at the left, work towards the right

[modest jasper]

I got the answer 😄

Thanks so much.

[vague belfry]

I have a linear transformation that is a reflection on plane x+2y+3z = 0

And I should determine the matrix of the transformation with respect to base v1=[1, 1, -1], v2 = [-1, 2, -1], v3 = [1, 2, 3]

The matrix of this transformation is
[1 0 0]
[0 1 0]
[0 0 -1]

correct?

Since v1 and v2 lay on the plane so they will not move, and v3 is perpendicular to the plane so the coordinates will become negative

now I have a problem determining the matrix with respect to the standard base

how do I do it xD

[slow scroll]

the matrix you wrote is for a reflection over the xy plane, not the plane x+2y+3z = 0

oh wait i see what you mean

[vague belfry]

with base v1 v2 v3

I guess, v1 and v2 stay the same since they are already on the plane

yeah

[modest jasper]

Does anyone know how I can do this question?

[red night]

youre looking for a vector X, such as AX=0

[modest jasper]

How do I find that

[red night]

you could just solve the system

take the vector X=(a,b,c,d) vertically

and find a b c and d such as AX=0

Or you could just try to check what answer is correct

by just multiplying the vectors by A and see if it gives you 0

[tranquil ermine]

it's the fixed set of the matrix

Px = x

so the solutions to (P-I)x = 0

[low hornet]

lil eigenvalues

[tranquil ermine]

for any projection

a projection is actually defined by that property

that P^2 = P

so the elements in the image are fixed by P

and yeah you may find the direction of projection by the kernel of P

but in general this is more than 1 dimensional;

projection from R^3 onto a line for example has a 2 dimensional kernel

so it isn't as nice

[wanton glade]

Does a matrix A have the same eigenvalues as A transpose?

[slow scroll]

yea because the determinant of A - \lambda I is equal to the determinant of its transpose.

that works because the identity is symmetric

[wanton glade]

Awesome, thank you

[slow scroll]

np

[low hornet]

@wanton glade yes

[haughty rune]

I'm not sure how to go about answering this...

could someone point me in the right direction

[olive breach]

I am taking this course https://www.udemy.com/share/100dKwAkQZcVxaQw==/

[low hornet]

@olive breach that B = (X^TX)^-1X^Ty on the cover there is a called the least squares solution

of a matrix

[olive breach]

I program multiaxis cnc machines and I want to understand this http://linuxcnc.org/docs/devel/html/motion/5-axis-kinematics.html

[kind arrow]

can someone explain this property of the dot product? $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$

[stoic python]

Randomdude:

[kind arrow]

if you do @ me with the explanation

[red night]

why do you want an explanation?

its just a property

[onyx tundra]

it should be pretty obvious from either of the standard definitions of the dot product @kind arrow

if you have a dot b = |a||b| cos(theta), then (ka) dot b = |ka| |b| cos(theta) = |k| |a| |b| cos(theta)

[kind arrow]

oh

[onyx tundra]

if k is positive, then cos(theta) doesn't change because a and ka are parallel

[kind arrow]

wow I am such an idiot thank you

yeah I just looked at it but I didn't make sense

but that explains it

[onyx tundra]

if k is negative then theta becomes 180°-theta and cos(180°-theta) = -cos(theta), so since if k is negative |k| = -k, the negatives cancel out and you get k|a||b|cos(theta) in either case

[kind arrow]

ah ok thanks

[onyx tundra]

np

[grave plank]

what are people's opinions on Jim Hefferson's Linear Algebra (the free one) and Howard Anton and Chris Rorres, Elementary Linear Algebra 11th edition (2014)

[frank gate]

Find the vector x that is transposed on the equation seen on picture: https://gyazo.com/34afdc133fc4f6a8863b03127f6d9f4c

How do i do?

Is it just ( X_1 = -1, X_2 = 2) ?

[low hornet]

ye

[frosty granite]

ho

how

[wanton glade]

If two matrices have the same eigenvalues, will they have the same eigenvectors?

[dusky epoch]

no

very much no

[wanton glade]

I thought so, thank you

[summer dagger]

What does the notation W^⟂ mean?

[brittle juniper]

it's the orthogonal of W

[summer dagger]

so what does that mean exactly confused by what the textbook is saying

x is in W^⟂ is saying x is an orthogonal vector to W or in W?

[brittle juniper]

If you've got a vector space $E$ provided with a dot product $\brk{\cdot,\cdot}$, \
then $W^\bot=\brc{x\in E~|~\forall y\in W,~\brk{x, y}=0}$

[stoic python]

Tuong:

[summer dagger]

ty still kind of lost

[winter reef]

bascically set of vectors that are orthogonal to any vector of W

[brittle juniper]

every

[winter reef]

ye

[summer dagger]

but going back to the question i had earlier is x in W or x is in W^⟂ which is the set of all vector ⟂ to W?

x is in W^⟂ is saying x is an orthogonal vector to W or in W?

[brittle juniper]

if $x\in W^\bot$, then $\forall y\in W,~x$ and $y$ are orthogonal

[stoic python]

Tuong:

[summer dagger]

i don't get what the second statement is saying

all of y is in W

[winter reef]

For all y in W

x and y are orthogonal

[summer dagger]

alright ty I'm sure I'll be back soon

[brittle juniper]

👌

[hazy mica]

hello everyone

how do I prove that if A and B are similar matricies, then they have the same eigen value?

this isn't my homework or exam, I promise

[brittle juniper]

if A and B are similar, then exists an inversible matrix P such that A=P¯¹BP

take an eigenvalue λ of A
there exists a non null vector X such that AX=λX
so P¯¹BPX=λX
so BPX=λPX

P is inversible and X≠0 so PX≠0
so PX is an eigenvector for B with eigenvalue λ

In the same way you can show that if μ is an eigenvalue of B, then it's also an eigenvalue of A

[hazy mica]

oh wow

so simple

thank you very much

[brittle juniper]

You're welcome

[wanton glade]

If a matrix has the eigenvalue 0 is it diagonalizable?

[wet finch]

no

(0 1; 0 0)

[wanton glade]

Ok good lol

[winter reef]

Umm what about 1 0 first row, second 00?

[jagged pendant]

that's diagonal(izable)
but the question was is it necessarily diagonalizable, and the answer is no, not necessarily

buncho gave a counterexample

[frank gate]

https://gyazo.com/99a77301752a1310cfe120a07129418c

Does it matter what two points i take to get the normal vector? For example (p_1 - p_2) x (p_1 - p_3)

I got the normal vector to (2,-3,2) but the answer contains the normal vector (-2, 3, -2)

[dusky epoch]

normal vectors are defined only up to scaling factor

you're fine

[frank gate]

What do you mean by that? 😃

i am a newbie sorry

So my answer is correct?

[dusky epoch]

yes

[frank gate]

Oki awesome! ty

[broken hawk]

what that means is if you have a normal vector, then e.g. twice that (or in your case -1 times that) will also be one

because the normal vector is just some vector that lies in the normal line to the plane

as in, the line that is perpendicular to the plane

and doubling a vector on a line (nb: “line” in linalg means it goes through the origin!) just makes it twice as long, but still on the same line

[flint sequoia]

Hey, I have a couple quick questions, not in linalg rn but just learning about it on my own time.

Conceptually, why are affine spaces closed under subtraction?

I’ve heard it said that some basis B of V is always a linearly independent subset of a spanning set S over V. This seems wrong to me. My naive thought process is that there a) may be no linearly independent set of vectors defined by linear combinations of elements of S and thus obtainable by reduction of S, and b) that a basis defined by linear combinations of elements of S is not necessarily a direct subset of S. Meaning A+B, with A and B being members of S, might be an element of the basis, but my understanding is that it’s not part of a subset of {A,B}

[broken hawk]

before I say something really stupid, is an affine space just a different word for a vector space or is it something entirely different?

[winter reef]

Different

[broken hawk]

yea I just looked it up

[remote fable]

I'm learning about LU decomposition. For normal cases, all is good. What I don't understand is the degenerate cases like this : $ \begin{bmatrix}
1& 2 &3 \
0& 0 &1 \
0& 1 & 0
\end{bmatrix} $

[stoic python]

topbottom:

[broken hawk]

LU is a lie, it’s actually LPU where P is a permutation matrix

you have to exchange two rows here, which is what the P does

[remote fable]

Yeah but what is the exact algorithm? Suppose that we go from A and transform it to this matrix (let's call it U), and record the row operations in L, so we have A=LU (with U not in upper triangular form yet). now we need to swap 2 rows, right? Then we have PA= PLU. Now we need to somehow move that P to the right so we have L'(PU). But what happens to L'? How do we get from L to L'?

The textbook states this theorem: for any matrix A, we can decompose the matrix PA to LU, with P is the permutation matrix. But that requires we know the proper permutation a priori, which we do not. If we try and fail (like above), we have to start over again?

[summer dagger]

don't want to interrupt is it ok if i ask a quick question that is a one word answer?

if a matrix is Orthonormal is it invertible?

[broken hawk]

no
yes

answered both of your questions

[winter reef]

lol

[summer dagger]

ty

[remote fable]

A matrix is orthonormal iff $ QQ^{T}=I $ so it is invertible and the transpose is its inverse.

[stoic python]

topbottom:

[dusky epoch]

that's called orthogonal

[broken hawk]

yes, but it shouldn’t be

it’s inconsistent naming

imo, orthogonal matrices should be ones where $Q^TQ = \Lambda$, with $\Lambda$ diagonal and invertible

[stoic python]

Sascha Baer:

[broken hawk]

btw, imo, $Q^TQ$ is better notation than $QQ^T$. You may think it’s a technicality. however, in the QR-decomposition, if you decompose a nonsquare matrix, $Q$ may only have orthogonal columns, not rows

[stoic python]

Sascha Baer:

[broken hawk]

specifically, this happens

and if you use Q̃, then the rows are not orthogonal

depending on which algorithm you use, you may either get the first or the second decomposition

(gram-schmidt gives you the second, householder the first, for example. they’re equivalent)

[astral junco]

I'm a bit confused on this. how is v_1*i a member of R?

What does "R" mean in this context?

[devout moth]

a math term

idk lol

[broken hawk]

then don’t speak

[devout moth]

jeez don't have to be rude about it

[broken hawk]

basically, all you do is take $\mathbb{R}^2$ and identify the second coordinate with the imaginary part

[stoic python]

Sascha Baer:

[broken hawk]

so like… $$\begin{pmatrix} x \ y \end{pmatrix} \equiv x + iy$$

[stoic python]

Sascha Baer:

[broken hawk]

the $x,y$ parts are real numbers

[stoic python]

Sascha Baer:

[astral junco]

hmmm

so <x,y> would no longer be a member of R^2 right?

[broken hawk]

what are the <,>?

that usually means inner product, but that wouldn’t make sense here

[astral junco]

oh sry i just meant the x,y vector you wrote

sry about that i wazs trying to represent a vector. not sure if i did it right

[broken hawk]

the vector (x,y) is an element of R²
x+iy is an element of ℂ

you show that essentially, the two are the same thing

by showing that if you add two vectors in R² and then go to C, or if you first go to C and then add them, you get the same result

[astral junco]

interesting. i'll have to think about this.

[broken hawk]

it’s called an isomorphism

note that usually when we talk about ℂ, we mean ℝ² with the additional structure of multiplication

[astral junco]

i guess i'm just having trouble with what a vector space is. so C is the vector space that (x,y) lives in?

[broken hawk]

no. you have two spaces here

in the space ℝ², we have vectors of the form (x,y)

in the space ℂ, we have vectors of the form x+iy

what you show in this exercise is that the two are essentially the same if you consider them vector spaces over the field ℝ (this means that the operation “scalar multiplication” takes real numbers as its scalars)

a vector space is three things:
•a set of elements
•a rule to add elements
•a field of “scalars” together with a rule for scalar multiplication of a scalar with a vector

a field being… a number system essentially

one in which you can add, subtract, multiply and divide

like the real numbers, or the rational numbers, but not the integers (because you can’t divide)

now, let me just describe how all these things look in ℝ²

[astral junco]

gotcha that clarifies it in my mind a lot

[broken hawk]

the elements take the form $\begin{pmatrix} x \ y \end{pmatrix}$, where x and y are real numbers

[stoic python]

Sascha Baer:

[broken hawk]

the addition rule looks like this: $\begin{pmatrix} a \ b \end{pmatrix} + \begin{pmatrix} c \ d \end{pmatrix} := \begin{pmatrix} a+c \ b+d \end{pmatrix}$

[stoic python]

Sascha Baer:

[broken hawk]

the scalars are real numbers, and if $\lambda$ is some scalar, then the scalar multiplication rule looks like this:
$\lambda \begin{pmatrix} a \ b \end{pmatrix} := \begin{pmatrix} \lambda a \ \lambda b \end{pmatrix}$

[stoic python]

Sascha Baer:

[broken hawk]

I hope all of this looks familiar enough

[astral junco]

yep

[broken hawk]

now, how does this all look in ℂ?
Well, now the vectors have the form x + iy, where x and y are real numbers.
addition looks like this: (a + ib) + (c + id) = (a+c) + i(b+d)
and we can again choose the scalars to be real numbers, and then λ(a+ib) = (λa) + i(λb)

now if you compare this to the above, you can see that it’s essentially the same thing, except that instead of writing it in a column, we write it as a sum, with an “i” in front of the second element

so ℂ can be seen as a two-dimensional real vector space

the confusion arises (possibly) because ℂ is also itself a field, and you can therefore see it also as a one-dimensional complex vector space. and those are not the same thing anymore

[astral junco]

hmm yeah now that i see what it means it's pretty pretty straightforward

[broken hawk]

(also I said a bit of a half-truth above: the vector space isn’t just any thing that has those three things, they also have to satify some axioms)

[astral junco]

think i got tripped by the wording vector space "over" R. couldn't figure out what "over" meant. now that you've explained it i think i get it.

[broken hawk]

(which basically tell you that addition and multiplication work exactly how they should)

[astral junco]

yeah i see what you're saying. it's a bit abstract, but it makes sense.

[broken hawk]

also, imo the whole “vector space over a field” thing really only makes sense once you’ve seen some other interesting examples of fields

so uh… have you met $\mathbb{F}_2$ yet

[stoic python]

Sascha Baer:

[astral junco]

no what 's F_2

[broken hawk]

it’s the field with two elements. so, you have two numbers, 0 and 1 only

and the following rules:

0 + 0 = 0
0 + 1 = 1 + 0 = 1
1 + 1 = 0

for addition

0 · 0 = 0 · 1 = 1 · 0 = 0
1 · 1 = 1

for multiplication

if you stare at this hard enough, you may recognize the rules of odd and even numbers: odd + odd = even (1+1=0) and so on

[astral junco]

hmmm

[broken hawk]

I wrote the numbers in bold to remind you that they’re not the real numbers 0 and 1

[astral junco]

what are they?

[broken hawk]

they’re just the things above

[astral junco]

some representation of T and F

[devout moth]

hey guys

how has your day been

[astral junco]

oh

[broken hawk]

ongoing discussion already, go somewhere else please, thanks

now, since this is a field, you can have vector spaces over 𝔽₂

[astral junco]

yeah there's a chill section guy. i'm trying to learn here

[broken hawk]

so for example, $\mathbb{F}_2^2$ would be vectors of the form $\begin{pmatrix} x \ y \end{pmatrix}$, where x and y are elements of $\mathbb{F}_2$

[stoic python]

Sascha Baer:

[broken hawk]

basically the same thing as ℝ², but with elements from a different field

there are only four elements in 𝔽₂², namely:

[stoic python]

Sascha Baer:

[broken hawk]

and you can think about what adding those would look like

[astral junco]

hmmm so is this like a seperate kind of number system? 1 and 0 are not real numbers?

[broken hawk]

exactly

a field is just any number system where you can add, multiply, subtract and, notably, also divide through anything except for 0; and the operations satisfy what you might call the rules of arithmetic

things like commutativity and distribution laws

[astral junco]

i've got a question about vector spaces which are kind of like these fields.

the set of rational numbers is not a vector space over R because if x,y and k are members of R then not every combination x+y*k is a rational number?

like in the previous example with complex numbers every x+yi represented a complex number but here not all of them represent a real number?

does that make any sense?

[broken hawk]

this is correct

you can also just use a counterexample

take the rational number 1

[broken hawk]

as a vector

and the real number π as a scalar

then π*1 would have to be a vector, ie rational

but it clearly is not

…actually use √2, just in case the person who corrects it gets all annoying and asks you to prove that pi is irrational

:P

[astral junco]

haha yeah i'm not sure how i would even do that

[broken hawk]

uh I think it’s doable but not easy

[astral junco]

thankfully no one will be correcting it. believe it or not, i just finished a linear algebra college class, but i think i have to start over from the beginning because clearly i did not absorb all the information lol

[broken hawk]

tbf, if it wasn’t a theoretical class this stuff was probably rushed

[astral junco]

yeah the class description said it would focus on applications of lin alg. idk it took me forever to learn calculus also so it might just be i'm not really a "math person". i'm kinda a slow learner. i do like math a lot though.

[broken hawk]

hey you could follow my dry af explanation

that says something

[red night]

in the culture part of my course i have this proposition: let A be a ring
M in Mn(A) is invertible <=> det(M) is invertible in A

is that true?

because det is a morphism of Mn(A),× to A,× such as det(Id)=1 right? So if M is invertible then the det is too

[fringe cave]

I'm p sure that's right

since if det(M) is 0, then it isn't invertible

[red night]

If that's true it would be beautiful

yes in a field

[fringe cave]

I don't know about the Ring one, but in a field yeah

Didn't see the ring part

[red night]

i know that's true in a field

the question was: Is it also in a ring?

if it is it's a beautiful result

[fringe cave]

B u t, it would certainly make sense given how matrices work, I'd have to actually look at it

[broken hawk]

I vaguely recall this to be true. iirc you can prove it with cramer's rule

[red night]

I don't know Cramer's rule

i go check it

[fringe cave]

^ yeah, it should be true, but i can't be bothered to actually prove it

[broken hawk]

3b1b has a video on it

it's a kinda ugly but occasionally useful rule for determining inverses

[fringe cave]

It's rather ugly i must say

[broken hawk]

yea. never use it in a calculation, only in proofs

[fringe cave]

if you use it in calculation, kindly uncalculate please

[red night]

in my course its said that if f is a function of A into B such as f(ab)=f(a)f(b) and f(1)=1 then f(A*) is included in B*

so it should be true i think

so i just need to show that if det(M) is in A* then M is invertible in Mn(A)

[dire hound]

hey

if I try to reduce a matrix to RREF and an entire row ends up being zeros

do I just take that row out and make the entire matrix one row shorter?

[dire hound]

please

[astral junco]

i'm not sure what you mean by taking the row out?

what are you trying to do exactly

@dire hound

did i prove that the set is not a vector space correctly?

[dire hound]

actually let me rephrase my question

what happens when one row of a matrix is a scalar multiple of another?

[astral junco]

well it means that you have free variables in your solution set for one

i'm pretty sure

are you trying to solve a system of equations?

[dire hound]

yes

[astral junco]

ok so do you know what a pivot column is?

[summer dagger]

feel like it's late for me to be asking this but does anyone know a good way or video to visualize row , col and nul of a matrix

[astral junco]

if type "3blue1brown null space" into youtube he has a great visual explanation of it

[summer dagger]

ik it's just a shame he doesn't have one on orthogonality cause then I'm lost

I'm trying to find something showing rowa perp = Nul a

[astral junco]

yeah beyond 3b1b videos i'm not sure. i'm retaking lin alg from the beginning because i'm lost as well

[summer dagger]

have a final tomorrow and every hour I study i feel like ik less lol

[dire hound]

why can there be multiple bases for a vector space?

[astral junco]

there are

at least sometimes there are infinitely many

i think

[dire hound]

thx

[half ice]

But all basis will have the same number of elements

Ergo vector spaces have a dimension

There's infinitely many

[astral junco]

my books says the answer is that it's not a vector space but for a different reason.

just trying to see mine is correct as well.

[slender yarrow]

works as well

[astral junco]

ok thank you 👍🏼

[slim pebble]

hey guys if a matrix is 5x5... would this be NO for diagonizable?

68.64208 + 0.00000i -3.64208 + 0.00000i -0.00000 + 0.00000i -0.00000 - 0.00000i -0.00000 + 0.00000i

these are the eigen values

[summer dagger]

dont think it would be cause doesnt have 5 distinct values but I'm not the person to answer lol

[slim pebble]

that what i was thinking too tbh lol

[sonic osprey]

You can't tell, it could be or it could not be diagonalizable

The identity matrix doesn't have two distinct eigenvalues, but is diagonalizable

[slim pebble]

this is the matrix itself

http://prntscr.com/noo3vb

[sonic osprey]

Review the definitions of algebraic multiplicity and geometric multiplicity

[stoic python]

woopy:

[slow scroll]

yea

[modest jasper]

Does anyone know how I could solve this?

[rigid cypress]

what it's asking for is two non-trivial solutions to the matrix

@modest jasper

so for the matrix A, find x1 and x2 for which Ax1 = 0 and Ax2 = 0

[modest jasper]

What does it mean for the matrix to be equal to something?

[broken hawk]

that’s not what it’s asking

for a matrix to equal something just means they’re the exact same matrix. but it’s asking you to find a product of matrices here which will result in 0

you have an equation of matrices and have to solve for x, where x is a matrix :P

and it’s specifically asking for a nonzero matrix, so your answer is wrong because of that

[frank gate]

https://gyazo.com/3340e466e7fc370a92dd894a6c9f5df1

Im on exercise 4

I got it to 3(x-1)+1(y-2)-2(z-3) = 0

Is this wrong?

[dusky epoch]

no, this is correct

[frank gate]

hmm okay 😮

the solution sheet has (3z + y -2z = 3+2-6

or (3x +y -2z = -1)

[dusky epoch]

that matches what you have

[frank gate]

oh oki 👌

[wintry steppe]

where'd it go

was reading it

[frank gate]

I think i figure it out! Gonna try

[wintry steppe]

Normal vector is orthgonal when you cross

ok ping me if u need anything

[frank gate]

yep got it

I am in need to find the equation of a plane passing through the line of intersection between two planes and a point

What is the strategy to finding the normal vector for this type of example?

https://gyazo.com/c5164f41bebbc83352fd2a81e61b6e24

im on exercise 8

[wintry steppe]

So remember that the coefficients of a plane are the direction vector (if I remember correctly?)

lemme get my notebook

yeah its in general form

so whats the normal vector of the plane of 8?

@frank gate

[frank gate]

is it the (plane 1 vector - plane 2 vector) ?

cross product i meant

(2, 3, -1) x (1,-4,2)

[wintry steppe]

yea that looks right

err

AB X AC?

=n

i think

[frank gate]

yes

and AB, AC are already calculated from the two plane equations given?

[wintry steppe]

Well you could find any point in that plane

suppose x and y = 0

solve for z and you get point a

[frank gate]

How do i solve for z in that case? I am awful at this

[wintry steppe]

https://i.gyazo.com/thumb/1200/c5164f41bebbc83352fd2a81e61b6e24-png.jpg

x=y=0

then -z=0

z=0

for that piont

and the other plane x-4y+2z=5

x=y=0

2z=5

[frank gate]

oki 😮

[wintry steppe]

2z=5

[frank gate]

ye

z = 5/2

z = 5/2

[wintry steppe]

so point A lines on plane 1 and it is (0,0,0)

and point B on plane 2 is (0,0,5/2)

[frank gate]

Ye the answer is: 5x-9y+5z = -15

i got to: 2(x+2)-5(y-0)-1(z+1) = 0

[wintry steppe]

try your other point

what you get

[frank gate]

https://gyazo.com/d4601afcc236535802d5fc8f74574a17

Didn't try with that. The solution sheet says like this. I still dont get that

[wintry steppe]

let me check something

[frank gate]

oh wait

[wintry steppe]

,w cross (2,3,-1) and (1,-4,2)

[stoic python]

Display results online and refine query

Requested by Babowwz#1394

[frank gate]

yes

that is what i got

[wintry steppe]

well well well

point is 0,0,0 right

and 0,0,5/2

point in reference was -2,0,-1

ab = (-2,0,-1)
ac= (-2,0,-7/5)

,w cross (-2,0,-1) and (-2,0,-(7/5))

[stoic python]

Display results online and refine query

Requested by Babowwz#1394

[wintry steppe]

no

i feel like im forgetting something but my notbook

[frank gate]

I think you are suppose to combine the planes and introduce a variable F for example and then that equation should correspond to the point of refrence you had

check this e.x: https://gyazo.com/b2df64af6d92a077ba859fb68feead82

same question, different values

[wintry steppe]

I haven't dealt with the lambda behaviors yet

[frank gate]

just think of it as variable F instead

[wintry steppe]

ay fren @half ice or @wooden plover you bois know bout this?

[frank gate]

so for example, x + F(2x) = -2

is the equation for x i believe

? 😮

[wintry steppe]

https://i.gyazo.com/thumb/1200/c5164f41bebbc83352fd2a81e61b6e24-png.jpg

number 8

[frank gate]

https://gyazo.com/1c3d60cddce0d3bfe2684a00c31da4ba

has this solution sheety

sheet*

i think i understand now

[wintry steppe]

<@&286206848099549185>

[frank gate]

you combine the two planes, insert x, y, z of the point on all the positions and solve for introduced variable F

[wintry steppe]

Yea, theres a couple m8s who know this just wait for someone to roll aroudn and check on it, ping me when u got a method

[frank gate]

hihi oki!

Ye that is what you gotta do, so i know it know

[half ice]

Wat? Are we all good now?

[frank gate]

i think so : )

[frank gate]

https://gyazo.com/f617fb3eb242165cdc0fb82fee47c9c1

im on exercise 9. Can someone explain what to do?

this is the solution sheet but i dont get it

https://gyazo.com/ba8ee890001fb7139c36c1f209c1cfa9

where the heck does x = -1 + 2t come from for example?

[half ice]

Lel essex. I had his class in PDE, interesting guy

[frank gate]

? 😮

[half ice]

I don't understand the solution either. That's an equation for a line, no?

[frank gate]

Ye read the top of this https://gyazo.com/f617fb3eb242165cdc0fb82fee47c9c1

and im on 9

[modest jasper]

@broken hawk Do you know how I might be able to solve the question then?

[broken hawk]

why are you pinging me

[modest jasper]

You responded to me...

[broken hawk]

sorry, I didn’t recognize you

[wintry steppe]

https://discordapp.com/channels/268882317391429632/540211747613704221/578128992654327809

[broken hawk]

yes, I do know how to solve it, but the person with the green name already told you what to do

and I don’t believe spelling solutions out explicitly is helpful

you should review your material on how to solve an equation like Ax = 0 (where A is a matrix and x, 0 are vectors)

and then, as was said there, your target matrix will just have (any) two solutions for that equation in it

[modest jasper]

I thought you said that's not what it's asking.

[broken hawk]

there’s infinitely many of them

then you misunderstood what I said

read it carefully

[modest jasper]

Okay.

[broken hawk]

it’s telling you that the solution itself may not be 0. not the righthand-side, but the thing you’re solving for

[modest jasper]

Alright, thanks.

[dusky mesa]

im trying to find a generalization for multiplication to the power of n for a 2x2 matrix, this is doable right?

i dont want to waste hours

im using eigenbases

[cloud cedar]

if you have an orthonormal basis of eigenvectors then taking A^n just means (a_{ii})^n

since it’s diagonalized

[broken hawk]

if you’re in the eigenbasis then it’s just taking the power of the diagonal entries

[astral junco]

I don't understand how S = {1} is a vector space. I thought vector spaces must have a zero vector?

[slow scroll]

it looks like addition is defined differently on this set. 1 acts as the zero vector

[astral junco]

ah good point

[slow scroll]

let $v \in S$. Vector addition on this set says that $v+1 = v$, so $1$ is the additive identity. Similar thing for scalar multiplication

[stoic python]

kxrider:

[astral junco]

interesting. so it becomes a trivial space under these operations i guess

[feral crow]

What do the B's notate

in part d?

$-B_x ; -B_z$

[stoic python]

Cosack:

[feral crow]

Or do I just treat them as variables

[cloud cedar]

looks like they're intended to be constants; one for the x direction and one for the z directoin

weird that they capitalized them though

@feral crow

[feral crow]

How do I find a vector that diagonalizes H?

I know A = PDP^(-1)

Is that asking for D?

Or P?

[cloud cedar]

you know what D is already once you find the eigenvectors/values of H

writing H in the basis of eigenvectors diagonalizes it; the diagonal entries are the eigenvalues

it's asking for P

[feral crow]

So P is just the eigenvectors?

In matrix form

and D has the cooresponding eigenvalues in the diagonal terms

[cloud cedar]

do you know braket notation? it makes it a bit more clear imo

[feral crow]

$P = [v_1 v_2]$

[slow scroll]

$\bra$

[stoic python]

kxrider:
Compile Error! Click the reaction for details. (You may edit your message)

Cosack:

[cloud cedar]

$\mel{i}{H}{j} = \sum_{n,m} \braket{i}{v_n}\mel{v_n}{H}{v_m}\braket{v_m}{j}$ where $i$ and $j$ range over the original $x/y/z$ basis and the $v$s are the eigenvectors. since they form a complete set, the sum over the projection operators $\sum_n \outerproduct{v_n}{v_n} = 1$ and you can insert the sum over eigenvectors wherever you like

[cloud cedar]

but basically yes, you write the eigenvectors as a matrix in the original xyz basis to get your rotation matrix

if this doesnt make sense to you, you can just ignore it (im a physics person so seeing the dirac notation is more illuminating for me, but it might not be the case for you)

[feral crow]

So when I use this calculator

D mages sense because it's just diagonal eigenvectors

But wtf is going on in the denominator

Do they normalize or something

[cloud cedar]

you can multiply v1 and v2 by x without changing their eigenvector status

it will probably make the math a little prettier

[feral crow]

Well the first column of P is clearly v_1

[cloud cedar]

since x is a constant, $\hat{H}v_1 = \lambda_1 v_1$, you know $\hat{H}(x v_1) = x\hat{H}v_1 = x\lambda_1 v_1 = \lambda_1(x v_1)$

[feral crow]

I just dont know why they have the denom

[cloud cedar]

ohh i see

[feral crow]

Maybe just how the calculator works

[cloud cedar]

yeah that looks like a normalization thing

the rotation matrix has to be unitary

[feral crow]

Since they all have the same term I can just ignore it i suppose

That denominator is definitely out of the scope of my class

[cloud cedar]

you can write it in wolfram and see if it simplifies

[feral crow]

Gotcah

Gotcha*

Thanks for the help 😃

[cloud cedar]

no prob!

[modest jasper]

Does anyone know how I can solve a question such as this...?:

[slow scroll]

yes

think about the column by coordinate definition of matrix multiplication with vectors.

[modest jasper]

I don't really know what that is, but I'll look it up.

Thanks!!

[slow scroll]

if you have a matrix with columns, $v_1, v_2, v_3, $ then $$[v_1 \quad v_2 \quad v_3] \begin{bmatrix} a \ b \ c \end{bmatrix} = av_1 + bv_2 + cv_3$$. This is completely equivalent to the other definition where you multiply the entries of each row with each corresponding column and add them together. @modest jasper

[stoic python]

kxrider:

[modest jasper]

@slow scroll Oh, yes, I figured that out.

I just want to know how I can determine what that matrix actually is.

Wait, I figured it out.

The answer was actually pretty simple.

Thanks again!!!

[slow scroll]

nice. no problem

[feral crow]

Any advice on starting this derivation?

[cloud cedar]

it's not actually as bad as it looks if you remember the multivariable chain rule; you have $x$ as a function of $r$, $\theta$, and $\phi$, so you can expand like $$\dd{x} = \pdv{x}{r}\dd{r} + \pdv{x}{\theta}\dd{\theta} + \pdv{x}{\phi} \dd{\phi}$$

[cloud cedar]

and similarly for dy and dz

[feral crow]

Is the determinant of a rotation matrix always 1?

Since that means it preserves the dimensions

er volume

i guess

[dusky epoch]

yes

proper rotations have det 1

[feral crow]

And thanks stellar!

[dusky epoch]

in fact rotation matrices satisfy a stronger condition

they're orthogonal

meaning $A^TA = AA^T = I$

[stoic python]

Ann:

[feral crow]

Thanks!

[feral crow]

How are the two equal?

I don't see that at all

[stoic python]

Cosack:

[feral crow]

I get a SUPER messy set of equations when I square dx, dy, dz

[modest jasper]

Does anyone know how I can solve a question like this?

[feral crow]

My best guess is C and D aren't linear because they move as a function of x in two different dimensions

So maybe that makes them not linear

But I don't know for sure

[modest jasper]

Oh, okay.

Removing them doesn't work unfortunately.

[sonic osprey]

What does it mean for a transformation to be linear?

There are two rules it needs to follow

[modest jasper]

Being honest I'm not exactly sure.

I did a lot of reading but it's still a bit hazy to me.

I know about x+y/3x.

[hoary nimbus]

Definition of linear to most people :
T(av + bw) = aTv + bTw

For a and b in K
For v and w are vectors

[dusky epoch]

that is the definition of linearity yes

[modest jasper]

I'm still not sure how that helps, I'm sorry....

[hoary nimbus]

Well added a disclaimer in case there are other notions

[dusky epoch]

you can separate it out into T(v+w) = Tv + Tw and T(cv) = cTv

[modest jasper]

I'm still not really sure how it helps, sorry.....

[hoary nimbus]

@modest jasper try substituting the vectors (x y z) with (x y z) + (x' y' z')

[modest jasper]

What is (x' y' z')?

@hoary nimbus

[hoary nimbus]

Just dummy you use to test

[modest jasper]

I'm still not sure how that helps, sorry...

I don't get why that makes a difference, I mean.

[hoary nimbus]

First try acting T on original vectors + dummy vectors then see if it equals T acting on original vector plus T acting on dummy vector separately

First property of linearity is just see if T(v + w) is it equal to Tv + Tw?

[modest jasper]

You mean T as in x+y/3x?

[hoary nimbus]

Where v = (x y z) and w= (x' y' z')

[modest jasper]

What does that have to do with the other side of the equations?

How does it factor in?

[hoary nimbus]

T is like some operation you do to vector v = (x y z) to produce things on the right

So T is not really numbers or anything (though it can be represented with matrix)

[modest jasper]

So it's not x+y/3x?

[hoary nimbus]

@modest jasper where do you get that x+y/3x from?

[modest jasper]

Khan Academy tbh.....

[broken hawk]

"it can be represented with a matrix" only if it's linear!

[hoary nimbus]

@broken hawk yeah almost forgot that

@modest jasper forget trying to factor T or making sense what number T is. T is just operation you do

To vector (x y z) and (x y)

[modest jasper]

So for something like the first question, would T be something like:

0 0

0 4

?

[hoary nimbus]

Seems correct

[modest jasper]

Okay, I'll be back in a second...

[broken hawk]

“forget trying to make sense what number T is.”
“so T would be (numbers)”
“seems good”

I hope you see the irony in this exchange

[hoary nimbus]

@broken hawk no comment

@modest jasper but you can NOT represent NON linear transformation with matrix. So the easiest check is, for example the first one is

try acting T ((x y)+(x' y')) = T(x+x' y+y') = (0 4(y+y'))

The outside parenthesis is the parenthesis for vector

[broken hawk]

this would be less confusing with latex

[wintry steppe]

what makes linear algebra difficult im thinking of trying to self study it

[modest jasper]

Self-study?

[broken hawk]

it’s not really difficult at all, tbh, but it tends to be somewhat of a spike in rigor

[hoary nimbus]

@broken hawk yeah typing latex in phone is hard

[broken hawk]

if you’ve never done any proof-based math that’ll be the challenge, mostly

[wintry steppe]

oh i c

[modest jasper]

@hoary nimbus I'm really sorry but what you said confused me a lot...

[wintry steppe]

how to get better at proof based math

[broken hawk]

do proof based math

[wintry steppe]

true

[hoary nimbus]

Haha sorry maybe @broken hawk can explain better

[broken hawk]

the issue with proofs however is that it’s easy to fool yourself into thinking you got sth right

[wintry steppe]

yeah

[broken hawk]

you have to get regular feedback on proofs you write

until you’re confident

[wintry steppe]

do u guys do that

like check proofs

[broken hawk]

if you don’t like spam it

sure

anyway, lemme see if I can explain this

lemme show you two examples in one dimension to illustrate how to check this

Example 1: $T(x) = 3x$

[stoic python]

Sascha Baer:

[broken hawk]

now you need to check if $T(x + y) = T(x) + T(y)$ and $T(\lambda x) = \lambda T(x)$

[stoic python]

Sascha Baer:

[broken hawk]

where x,y are vectors (which in 1D are just numbers) and lambda is a number

so let’s start at the beginning

[stoic python]

Sascha Baer:

Sascha Baer:

[broken hawk]

are they the same?

[modest jasper]

Yes?

[broken hawk]

good

what about the second property?

is $T(\lambda x) = \lambda T(x)$?

[stoic python]

Sascha Baer:

[modest jasper]

What about it?

[broken hawk]

does it hold?

[modest jasper]

Yes.

[broken hawk]

proof?

write it out please

[modest jasper]

I don't know. Didn't you just tell me that was the rule above?

[broken hawk]

no?

you need to read more carefully

I said we have to check two properties

the first we’ve checked

T(x+y) = T(x) + T(y)

but we’ve not yet checked if T(λx) = λT(x)

for a function to be linear, both of these must be true

and one does not necessarily imply the other

[modest jasper]

Uh...

[dusky epoch]

||if your base field is Z_p for p prime, it does||

[broken hawk]

all you have to do is plug in the definition of T(x)

that’s all you have to do

just apply the function to whatever it is applied to

[modest jasper]

Is it that because T(x)=3x+3y, if you take out three and have 3T(x)=x+y, that will be equal to T(x)=3x+3y?

[broken hawk]

T(x) is not 3x + 3y, what?

please look at the definition

[modest jasper]

I'm sorry, I'm really confused...

[broken hawk]

please work more carefully

[modest jasper]

I don't know how to determine this lambda stuff.

[broken hawk]

you just need to take it slow. I gave you a definition and all you have to do is apply it

what is T(x)?

[modest jasper]

3x?

[broken hawk]

what is λT(x)? (lambda is just a number)

[modest jasper]

Uhhh.

3lambdax?

[broken hawk]

now, what is T(λx)?

[modest jasper]

The same thing?

[broken hawk]

yes

λT(x) = λ3x = 3λx = T(λx)

and now we’ve shown that T is linear

[dusky epoch]

i feel like linalg isn't the best place to start out for a student who isn't familiar with proofs

[broken hawk]

worked for me

[hoary nimbus]

@dusky epoch what's your recommendation?

[broken hawk]

now second example

$T(x) = x^2$

[stoic python]

Sascha Baer:

[dusky epoch]

i would recommend giving How To Prove It by Velleman a read.

[broken hawk]

Saten, can you try to calculate T(x+y) and T(x) + T(y)?

[modest jasper]

T(x+y) I guess would be x^2+2xy+y^2, and T(x)+T(y) would be x^2+y^2?

[broken hawk]

yes, that’s correct

and it’s pretty easy to see that those two are not always the same

[modest jasper]

Yes.

[broken hawk]

therefore, this function is not linear

while the first one was

[modest jasper]

But how would you do it if T(x) was a vector instead of a scalar?

[broken hawk]

you do the same procedure, but instead of x and y you use two vectors

and the function will be applied to each component of the vector inside

[dusky epoch]

well no

[broken hawk]

it’s hard to word

[modest jasper]

I'm a bit confused. Could you give an example? I pretty much only understand examples tbh.

[broken hawk]

ann, can you take over? I have to start working on my own stuff

[dusky epoch]

sure, gimme a min

i'm gonna get some paper

so here's an example of a transformation whose inputs are vectors, in this case from R^2

@modest jasper do you want to try it for yourself? check whether this transformation is linear, and if it is, write a proof of its linearity

[modest jasper]

What I need to know is what to do if the output is a vector.

[dusky epoch]

there really is nothing special about that

you know how to add and scale vectors, do you not?

[modest jasper]

Yes.

[dusky epoch]

well ok let me give you another example

[modest jasper]

I was wondering because all the examples I got before had T(x) equal to a single value.

[dusky epoch]

this is not multi-valued either

there is only one output; it just happens to be a vector

well

a vector from a space that isn't R

[hoary nimbus]

@dusky epoch I think she might get it with R^2 - > R^2

[modest jasper]

That example seems to be okay, but I don't know how to check if it is linear...

[broken hawk]

just work through the steps, one by one

1. calculate T(v+w) and T(v)+T(w), where v and w are now vectors. If they’re not always the same, then it’s not linear
2. Calculate T(λv) and λ*T(v), where λ is a scalar. If they’re not always the same, then it’s not linear

if they’re always teh same, then it’s linear

that’s all you have to do

now just do it

step by step

[modest jasper]

I'll try...

Thanks..

[dusky epoch]

don't overthink it

[modest jasper]

Can T be a vector instead of a single scalar value?

[hoary nimbus]

T is abbreviations of transformation

[dusky epoch]

T isn't a vector or a scalar

T is a function

[modest jasper]

How do I know what the value of the function T is then?

[dusky epoch]

you will be given it.

[modest jasper]

In my question I wasn't given T though...

[hoary nimbus]

@modest jasper you were given the value of T

[modest jasper]

Is that vector value the function you were talking about?

[dusky epoch]

ok wait
what is your original question?

[modest jasper]

What do I do if T is a vector instead of a single scalar value?

[dusky epoch]

you mean if T is vector-valued? i.e. its outputs are vectors?

[modest jasper]

In @broken hawk 's examples he only gave one value as the value of T.

That's not what's in the question.

So I'm still confused about how to answer my questions...

[dusky epoch]

sorry but i really don't understand your confusion

checking a transformation for linearity is EXACTLY THE SAME whether its outputs are numbers or vectors

[hoary nimbus]

maybe should start with definition of function as certain subset of Cartesian product of X \times Y just to remove prejudice what are function

[broken hawk]

yes, I gave examples with one value to show the procedure

because the procedure is the exact same

you just… have to actually do it

you’re way overthinking it

just start writing

$T\left(\begin{pmatrix}x_1 \ x_2 \end{pmatrix} + \begin{pmatrix}y_1 \ y_2 \end{pmatrix}\right) = \dots$

[stoic python]

Sascha Baer:

[broken hawk]

simplify the expression inside first, then apply the function based on the instructions given

[modest jasper]

I'm sorry, but I'm just not sure how to do this with both an x and a y and a vector @broken hawk .

I just don't really know how to determine what T is and how to plug in the values into the function.

[broken hawk]

calculate T for 500 values until you understand exactly how to do that, then replace the values with variables

what T is is given to you

[modest jasper]

Where?

[broken hawk]

e.g. in Ann’s example

and in literally every other exercise

it always says exactly what T does

you just have to apply it

[modest jasper]

It just says what you get what you have T(x y).

But I don't know what to do when I have two variables.

[broken hawk]

I said “simplify the expression inside first”

[modest jasper]

I don't know how to do that...

[broken hawk]

you need the inside to be a single vector, aye

do you know how to add vectors

[modest jasper]

Yes.

[broken hawk]

then do it

[modest jasper]

What should I add?

What is inside?

[broken hawk]

I give up

sorry, but I cannot help you anymore

[modest jasper]

Okay.

[hoary nimbus]

[dusky epoch]

[modest jasper]

I just think it's kind of unreasonable for you to expect me to figure out on my own how to complete these questions when your only example was something that was far simpler than what I have to do now.

[broken hawk]

I mean I also shouldn’t be helping anyway, I have a measure theory chapter to get through

[hoary nimbus]

@modest jasper how about reviewing the material? Or just ask your teacher/professor?

[modest jasper]

It's due in a few hours.

I can't.

[hoary nimbus]

You know you guys in America got things called office hour right?

[modest jasper]

I'm Canadian but okay.

[hoary nimbus]

do it with your friends

[modest jasper]

I don't have any friends.

[hoary nimbus]

Make one or two soon

[modest jasper]

r/wowthanksimcured

[empty copper]

Imagine being able to make friends

Kek

Anyway gl Saten

[modest jasper]

Thanks.

[thick tiger]

can someone help verify

[sonic osprey]

What have you tried?

[thick tiger]

bringing it to reduced row echelon form

plugging in t = 0

both didnt work

[sonic osprey]

What does it mean for these to be linearly independent?

[thick tiger]

their superposition, when = 0, only is valid for when all constants are 0

[sonic osprey]

Right and you have to remember this is a superposition of functions. So the thing on the right, the = 0 is really the 0 function.

[thick tiger]

yes

[sonic osprey]

So it's true for all t

[thick tiger]

what is

that the functions are = 0 true for all t?

[sonic osprey]

That the superposition of the functions = 0 for all t

[thick tiger]

if they are linearly independent, yes

i have to prove they are

setting them equal to 0 doesnt do anything since its all e

and trying to do ln(e) doesnt work since ln(0) is undefined

though rn im trying to see what happens if i divide all by e^t

[sonic osprey]

You want to show that all your constants are 0. Plugging in t = 0 will give you a system of equations so that you can solve for these constants

[thick tiger]

plugging in t = 0 will just net me the matrices given

the [1 1 0] etc

im not sure if "verify" means it can be dependent or independent

because so far everything ive done has shown dependence

[sonic osprey]

Remember what superposition actually is. You're taking a linear combination of these three vectors. So you have three constants

These constants are what you need to show are 0

[thick tiger]

got it

thank you

is this correct?

looked it up more

if i get a 0=0 situation, it's infinitely many solutions

and if thats the case, it's not linearly independent

so

honestly i've tried multiple things

more than once i've gotten linearly dependent

[dusky epoch]

tbh isn't it kinda obvious these are LI

[thick tiger]

yes

but the wording of the question implies they should be independent

but

i dont know

i feel like the fact that theres that e^t bullshit going on, its affecting it in some way that makes it independent

[dusky epoch]

yes it does

the only way to make u e^2t + v e^4t ≡ 0, where u and v are vectors, is to make u = v = 0

[thick tiger]

dunno where you got 4t from

but

i cant just say that, can i?

i dont want to lose points by just explaining

[thick tiger]

Did I do this right

[dusky epoch]

no, B is supposed to be a 2 by 2 matrix

specifically the identity

[thick tiger]

okay

for this one again, this is it?

[wintry dirge]

How do I get from the first to the second step? Where does (2, 3, 4) come from?

[dusky epoch]

what is A?

@wintry dirge

[wintry dirge]

I'm not quite sure, it says to solve the equation system (first step)

And the second step is to be the solution

So I guess A is a matrix?

@dusky epoch

[dusky epoch]

i mean yes A is a matrix

i was asking you for its value

can you post the whole problem

[rigid cypress]

wait, you wouldnt need A to find the vector (x y z)^T

what it's saying is for some unknown matrix A, Ax = (1 0 1)

[wintry dirge]

The problem is just: Solve the following equation system

[rigid cypress]

and for the inverse A^-1*(1 0 1), you would get x

[dusky epoch]

@wintry dirge does the problem not say "A = ( ... ... ... )" elsewhere

[rigid cypress]

wait, is the question asking you to find x y z from the the values given?

is there any more information in the question

[wintry dirge]

The problem before on the sheet is to Invert A and to prove it

[dusky epoch]

oh THERE we go

so THAT'S what A is

[wintry dirge]

wow

I feel dumb

Thank you lol

[feral crow]

Any tips?

[wintry steppe]

I was shown this method for finding the determinant of a 3x3 matrix (denoted by "A" in this first screenshot). It is expressed as the sum of 3 ,2x2 determinants which have each been multiplied by a scalar. http://prntscr.com/npg0y0 . I am then shown this general way of calculating the determinant of a N x N matrix. http://prntscr.com/npgcdy If I have a 4x4 matrix, i am to calculate the value of 4, 3x3 "minor" determinants. To calculate the value of each of these 3x3 determinants in order to calculate the value of the 4x4 determinant, am I to follow the same method in the first screenshot but repeat the method 4 times?

The minor determinant for a cofactor within A is generated by eliminating every element in the same row and column as that cofactor

[winter reef]

If I have a 4x4 matrix, i am to calculate the value of 4, 3x3 "minor" determinants. To calculate the value of each of these 3x3 determinants in order to calculate the value of the 4x4 determinant, am I to follow the same method in the first screenshot but repeat the method 4 times?" Yes you can do that.

There's also rule of Sarrus on how to find determinant of a 3x3 matrix

[wintry steppe]

Ok good

[winter reef]

but what you sent is called Laplace expansion

[wintry steppe]

I see

These are just my online university e notes

[frank gate]

Show that the two planes, x - y = 3, and, x + y + z = 0 interseet, and find a vector v pararell to their line of intersection

How should i approach this?