More concretely, now: we know that every vector space over is free as a module over . That is, every vector space has a basis — a set of vectors so that every other vector can be uniquely written as an -linear combination of them — though a basis is far from unique. Just how nonunique it is will be one of our subjects going forward.
Now if we’ve got a linear transformation from one finite-dimensional vector space to another, and if we have a basis of and a basis of , we can use these to write the transformation in a particular form: as a matrix. Take the transformation and apply it to each basis element of to get vectors . These can be written uniquely as linear combinations
for certain . These coefficients, collected together, we call a matrix. They’re enough to calculate the value of the transformation on any vector , because we can write
We’re writing the indices of the components as superscripts here, just go with it. Then we can evaluate using linearity
So the coefficients defining the vector and the matrix coefficients together give us the coefficients defining the vector .
If we have another finite-dimensional vector space with basis and another transformation then we have another matrix
Now we can compose these two transformations and calculate the result on a basis element
This last quantity in parens is then the matrix of the composite transformation . Thus we can represent the operation of composition by this formula for matrix multiplication.