Today I learnt that a linear transformation is the character that drives a matrix. If you choose a basis for an input $latex {v_1,v_2, \ldots, v_n }&fg=000000$and you choose a basis for the output $latex {w_1,w_2,\ldots,w_m}&fg=000000$ and the transformation is from $latex {T : \mathbf R^3 \rightarrow \mathbf R^2}&fg=000000$. If you want to describe a transformation, you describe it using a matrix. The matrix will tell us what it does to each of the input basis. So,let’s say I choose an eigen vector basis. In that case, the transformation is a diagnol matrix.

I think I need to read Axler