Question

[Linear Algebra] Is a matrix a function?

Answer 1

So today, my professor said that a matrix of rows n, and cols m, has a Domain of R^m, and a Co-domain of R^n. And it's quite confusing.

Now, I get that if you MULTIPLY two matrices together, the matrix to the left EXPECTS to be multiplied with a matrix whose rows equals to the number LEFT MATRIX's cols.

The result of course would have the same number of rows, if it's less than the number of rows of the RIGHT MATRIX, I assume that the transformation collapsed some of its components........


but how does that make matrices functions???? As far as I see it, the FUNCTION is the operation of MULTIPLYING. this FUNCTION demands the RIGHT MATRIX's row count to equal the the LEFT MATRIX's col count.

Answer 2

are you referring to systems of equations?
a matrix is not a function, it is just an array of number, which can be represented as something else

Answer 3

Well; it depends on how you interpret a matrix. You can interpret a matrix as a function, actually, contrary to what @sooobored is saying. It's really a linear transformation if you think about it

Answer 4

A matrix that is n by m is really a linear mapping from R^m to R^n

Answer 5

Matrix multiplication corresponds to the COMPOSITION of linear mappings

Answer 6

or, the composition of functions.

Answer 7

\[\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\]
Yes just a matrix.

Answer 8

http://mathinsight.org/matrices_linear_transformations

Answer 9

I get that. I get that a matrix can be interpreted as a transformation of space, or unit vectors. I get that through a transformation matrix like this, any vector matrix from the original space state can be converted to what it is in the new, transformed space, since all you're doing is applying a linear combination to some new unit vectors, the ones of the new space.

But does that make it a function? Don't you utimately also still have to *multiply* any vector with this transformation matrix for it to "transform" ? It feels like it's encoding the transformation data, and that, can be applied.

I'm also aware that transformations can be compounded as well. You have unit vectors that were transformed. Now all you have to do is transform them again, just like normal vectors, and they've been then transformed two times through two rules.

But i'm still confused howa standalone matrix is a function. It feels like transformation data to me, something, that a function can use to transform vectors!!

Answer 10

What is a function but a rule of a mapping from one domain to another codomain? Why must we need variables when they are already implicitly known?

Answer 11

Yeah that's true, and a matrix does represent the new values for the unit vectors, but...
What exactly is doing the mapping? Isn't it the matrix multiplication function? Isn't that operation the one applying the transformation matrix onto the vector?

Though it's true that the matrix can be visualized as a transformation, I do look at it like that. So I guess my execution is what the difference between a rule, and an execution of a rule is.

Answer 12

my confusion*

Answer 13

Well, I guess you have to expand your vision a bit...
1 2
3 4
IS BY DEFINITION, the mapping from R^2 onto itself, that is
x+2y=x'
3x+4y=y'
The multiplication IS implicit.

Answer 14

seems like the fellows at Georgia Tech concur...
https://people.math.gatech.edu/~cain/notes/cal6.pdf

Essentially what we are getting at is that there are multiple ways to represent the same thing. We don't need to explicitly state that we're multiplying our matrix by our vector in the domain since it is understood.

Answer 15

hmm well a function by definition is a mapping like you said, so I guess if I have any kind of information that maps a domain into a co-domain, that'd make it a function?

also at the definition you just stated, is that saying that this transformation means that, whatever state a vector [x y]^T is at, this represents a transformation of it into some new [x' y']^T? (I'm using ^T to mean that this is vertical, I hope i'm using it right ahaha)

Answer 16

yes, but do keep in mind that the dimensions of vectors in the domain and codomain need not be equal. The way I see it, a matrix can do a lot of things, but in the context of linear algebra, it's a shorthand way to write a linear transformation that essentially maps one set of ordered tuples to another set of ordered tuples. The transformation need not be one-to-one or onto as you may know, but we get interesting properties when it is.

Answer 17

yeah I know, as long as the vector's number of components, and the number of columns, or unit vectors match, we can transform, even if we end up collapsing that space into a lower rank

Answer 18

but it's like you said, multiplication is implied. I was stuck on the part that, yeah, the matrix is a transformation rule. but it is being applied through the implied multiplication operation to the other vector. so rather than just looking at it as a mapping, i was stuck on what actually is performing the mapping, and referred to that as the funciton. but now that i think about it, it seems like the matrix is a transformation rule, and matrix multiplication is a rule that applies the transformation rule to a vector, or a set of vectors

maybe i'm just being too dynamic, and it seems too static to me but i can see it as a function by virtue of being a rule

Answer 19

and even multiplication really is just a rule or procedure that states what has to be done, so the only dynamic part is really just what is applying the math, so i see my misconception there

Answer 20

who is applying the math*

Answer 21

To be honest, I had the same problem as you... It was not until I learned about functions as cartesian product subsets that I began to realize that the only reason things didn't seem to click was because I wasn't relaxing my viewpoints and expanding them a bit... I had the same problem in algebra when I struggled to understand how the equation y=mx+b could be a line... It really is, if you let so and so be so...

Answer 22

hmmm, so what idea should i ease off of? that anything can be a function, even a number? is a number, a scalar, a function?

Answer 23

Well, I don't think I know enough to answer that question any better than you, to be honest. But I always think of a function as a map from a domain to a codomain. One set, to another set...

Multiplication is a function... from R^2 to R, if you think about it that way...

Answer 24

yeah, it is!! I think the misconception really was that I thought a function is an action, rather than a recipe. So now that someone pointed at something that totally "looks like a recipe" and said it was an action i got all confused. cuz it feels too static

Answer 25

but it makes sense now

Answer 26

they never said it was an action, i only assumed it was ahahah