Matrices as transformations and column combinations

Columns tell you where the axes go

Suppose a matrix has columns $u$ and $v$:

Apply it to the first coordinate direction, $(1,0)$, and you get $u$. Apply it to $(0,1)$, and you get $v$. The columns record where these two basic directions land.

Build every other output from those two arrows

The input $(2,3)$ means two steps along the first direction and three along the second. Linearity sends it to two copies of $u$ plus three copies of $v$:

This is the column-combination view of matrix multiplication. The input coordinates supply coefficients; the matrix columns supply the arrows to combine.

Connect it to row–column arithmetic

The first output coordinate is $1\times2+0\times3=2$. The second is $1\times2+1\times3=5$. We have obtained the same answer by organizing the arithmetic differently.

The geometric view explains the whole transformation. Here the first axis tilts upward while the second stays fixed: the grid shears. A straight line of inputs still maps to a line or a point, and the origin stays at the origin.

Choose the coefficients

Which input returns exactly $v$?

Answer: $(0,1)$, because $0u+1v=v$. This is why the second column describes the image of the second coordinate direction.