I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example?

The natural motivation of matrices is as representing systems of equations.

If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space.

If I write a matrix, say, this:

  [[1  2]
   [3  4]
   [5  6]]

What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first vector space are represented as a linear combination of the basis vectors of the second vector space. Of course, the transformed vectors may not necessarily be a basis of the latter vector space.

> The natural motivation of matrices is as representing systems of equations.

That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.

Doesn't make sense is too strong though.

If you have a system Ax=y and a system By=z there exists a system (BA)x=z

This system BA is naturally seen as the composition of both systems of equations

And the multiplication rule expresses the way to construct the new systems' coefficients over x constrained by z.

The C_i equation has coefficients which are the evaluations of the B_i equation over the A_k-th coefficients

C_ik = B_ij A_jk

concretely

        A11 x1 + A12 x2 = y1
        A21 x1 + A22 x2 = y2

        and

        B11 y1 + B12 y2 = z1
        B21 y1 + B22 y2 = z2

        then

        B11 (A11 x1 + A12 x2) + B12 (A21 x1 + A22 x2) = z1
        B21 (A11 x1 + A12 x2) + B22 (A21 x1 + A22 x2) = z2

        rearrange and collect terms

        (B11 A11 + B12 A21) x1 + (B11 A12 + B12 A22) x2 = z1
        (B21 A11 + B22 A21) x1 + (B21 A12 + B22 A22) x2 = z2

the coefficients express the dot product rule directly

There's no single best way to understand any of this, but the action of a matrix on the standard basis vectors is a totally reasonable place to start because of its simplicity, and then the action on any vector can be built out of that because they're linear combinations of basis vectors.

i don't agree because this seems circular. You cant even define a matrix as something that acts on vectors meaningfully until you have some machinery.

if you start with a set S and then make it vector space V over field K. Then by definition, linear combinations (and its not an algebra so nonlinear isn't even defined) are closed in V.

You can then define spanning sets and linear independence to get bases. From bases you can define coordinate vectors over K^n as isomorphic to V. Then given some linear function f : V->W by definition f(v) = f(v^i * b_i) = v^i * f(b_i)

Only here is when you can even define a matrix meaningfully as a tuple of coordinate vectors which are the image of some basis vectors.

Then you need to prove that what was function application of linear functions on vectors is the same as a new operation of multiplication of matrices with coordinate vectors.

And then to prove the multiplication rule (which is inherently coordinate based) you are going make the same argument I made in sibling comment. But I could prove the rule directly by substitution using only systems of linear equations as the starting point.

Where's the circularity?

What you're saying is fine as an abstract presentation, but I was talking about how students might initially come to learn about matrices, so just introducing column vectors as representing points in 2 and 3 dimensional space and how matrices transform them is fine.

Beginning with the field and vector space axioms might be fine for sophisticated students, but I don't think it would make for an optimal learning experience for most students. We also don't teach kids the Peano axioms before they learn to add and multiply

But the question was about deriving the multiplication rule. I said you could derive it from systems of equations directly and gave a proof.

> and how matrices transform them is fine

this is circular. You are introducing/assuming the multiplication rule right here. You can't then derive it

Yes, I was assuming it, not deriving it, so there's no circularity.

I realise though that I was answering more "How does it work?" (application) rather than "Why does it work?" (derivation)

For the latter, something involving sets of linear equations is probably best, as you initially said