The Columns In Representation Of Linear Transformation As Matrix

Question

Why do we write the coordinates vector in the process of finding the representation of a linear transformation as matrix in columns?

Answer 1

If you want to represent the linear transformation as a matrix-vector product, writing them as columns have the representation of

$$Ae_i=A_i$$

where $e_i$ is the $i$-th standard unit vector and $A_i$ is the $i$-th column of the matrix $A$.

Answer 2

It always helps doing a (2 by 2) example.

Note what happens when we take $$ \begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}= $$ it spits out the first column of the matrix. I.e. the first column is the output of the linear map associated to this matrix, call it $T$, when applied to $T(1,0)$.

Answer 3

This is a convention that’s consistent with left-multiplication by a matrix to implement application of a linear transformation. You could write the vectors as rows instead, but then you’ll instead have to right-multiply by a matrix to match. (The appropriate matrix will be the transpose of what you’re used to.) This post-multiplication is (or, at least, used to be) a common convention in computer graphics.