Linear Transformation from R^m to R^n: Mapping Scalars to Vectors

[dumbQuestion]

Can we think of a linear transformation from R^m-->R^n as mapping scalars to vectors?

Let me say what I mean. Say we have some linear transformation L from R^m to R^n which can be represented by a matrix as follows:

L=[ a₁₁x₁+a₁₂x₂+...+a_1mx _m
a₂₁x₁+...
.
.
.
a_nmx₁+...+ a_nmx_m ](sorry for that ridiculous representation, just wasn't sure how to write it in this forum). Anyway its supposed to be the general representation of some nXm matrix.So this takes as imputs scalars (x1,x2,...,xn) and gives as output:

L(x1,x2,...,xn)=(a₁₁x₁+a₁₂x₂+...+a_1nx _n, a₂₁x₁+a₂₂x₂+...+ a_2nx_n,...,a_n1x₁+...+a_nmx_m)isn't this just like saying:

L(x1,x2,...,xn)=(a₁₁x₁+a₁₂x₂+...+a_1nx_n)i + (a₂₁x₁+a₂₂x₂+...+a_2nx_n)j,+...+,(a_n1x₁+...+a_nmx_m))t *(not sure what standard vector you would use if its n dimensional, i only know i j and k so i just randomly chose the letter t)So can't we think of the transformation like a vector field? Isn't this what the gradient ∇ does? (takes a scalar field and transforms it to a vector field) Now, we can't take the gradient of a vector field right, so how can we go about finding max and min points then? Because they occur when the gradient is 0 but how can I really think of the gradient of this?

 

[HallsofIvy]

You could, but if you do that, interpreting the separate components of the vector in Rⁿ as individual numbers, and the components in R^m as vectors, you lose important information- especially if you want to consider changing coordinate systems.

Better to think of the function as a matrix multiplication: y= Mx where, because x has n rows and y has m columns, M is a matrix with n columns and m rows.

 

[dumbQuestion]

But what if my goal is to learn things about say, the space this is mapped to. For example, say I have for example a closed bounded subset in R^m, O, and I want to look at L(O) (the transformation of O into R^n using my linear transformation L). Say I want to learn things about that range. For example min and max properties. I'm just kind of confused how I would say, take the gradient of that space to find that stuff out, or get to know properties about what's going on on the boundary. An example being say I have a transformation T that's mapping points in R^3 to points in R^2. So this can be represented as a 2X3 matrix. Now say I just want to evaluate T(C) where C is, say, a closed ball of radius epsilon. Now I want to know things about the output of this, like the min and max that we find in T(C).Is this a linear algebra question? Or a calculus question? Or a topology question? I'm just not sure where to look to study and understand this or maybe what theorems I should look at to understand it more?

 

[HallsofIvy]

Then you don't want to think of mapping numbers to vectors, you want to map points to vectors. And you can do that by assuming that each vector starts at the origin and then taking each vector to indicate the point at its tip.