On: Free Linear Algebra textbook

[jfarmer]

There are two relevant facts:

    1. Linear maps are determined by their behavior on basis elements
    2. The composition of linear maps is still linear

Together, this means you can work out how the composition of two linear maps affects the coefficients of each vector (relative to a fixed basis). If you do this, you get the "formula" for matrix multiplication.

Let V be an n-dimensional vector space. Let {e_1, e_2, e_3, ..., e_n} be a basis for V.

That means any v in V can be written

    v = a_1*e_1 + a_2*e_2 + ... + a_n*e_n

Let f: V → V be a linear map. Let's see what it means to apply f to v. We have

    f(v) = f(a_1*e_1 + a_2*e_2 + ... + a_n*e_n)
         = f(a_1*e_1) + f(a_2*e_2) + ... + f(a_n*e_n)
         = a_1*f(e_1) + a_2*f(e_2) + ... + a_n*f(e_n)

In other words, if we know the values of f(e_1), f(e_2), ..., f(e_n) then we can calculate the value of f(v) for any v.

Every choice of value for f(e_1), f(e_2), ... is valid and determines a unique linear map.

For n=2, use the standard basis where e_1 is (1,0) and e_2 is (0,1).

Write f(1,0) as (a,c) and write f(0,1) as (b,d). This is just "relabeling" the values f(1,0) — under the standard basis we know that f(1,0) looks like (a,c) for some values of (a,c).

Write v = (x,y) so that

    v = (x, y)
      = x*(1,0) + y*(0,1)
      = x*e_1 + y*e_2

Then we have:

    f(v) = f(x, y)
         = x*f(e_1) + y*f(e_2)
         = x*f(1,0) + y*f(0,1)
         = x*(a,c) + y*(b,d)
         = (x*a, x*c) + (y*b, y*d)
         = (x*a + y*b, x*c + y*d)

Or written using matrix notation:

    [ a b ] [x]   [x*a + y*b]
    [ c d ] [y] = [x*c + y*d]

So that's the "formula" for applying a matrix to a single vector. It's determined entirely by the four values (a,b,c,d), but really it's determined by the action of the map on the basis vectors.

Now let f,g be linear maps and calculate g(f(v)) in the same way. You'll get the "formula" for matrix multiplication.

In other words, matrix notation, matrix multiplication formulas, etc. are "just" compact ways of representing the behavior of linear maps.