Why are vectors vertical?

My math professors in college would often intimidate us into writing vectors as vertical, transposing them when convenient, without explaining why. This always confused me, but I realized that rather than being some Big-V conspiracy, they never explained it because there was a very obvious reason why.

An illuminating example

For example, take some transformation (ie, a matrix) $\mathbf{A}$ and a vector $\vec{x}$ on which it acts on.

Note that we format this function $\mathbf{A}$ as the hstack (horizontal-stack) of its vertical basis vectors in the transformed space. For now, suppose this is a good idea.

Then we can write

\[\mathbf{A} \vec{x} = \begin{bmatrix} u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}\]

to denote our equation.

Why are basis vectors of a matrix vertical?

Now the reason why we chose to format $\mathbf{A}$’s basis vectors along its columns becomes clear.

Now when we multiply it out, we get

• the $x$’s always multiplying with the $u_i$ components,
• $y$’s always multiplying with $v_i$ components,
• and $z$’s always multiplying with $w_i$ component.

So the purpose of aligning the matrix along the columns

\[\begin{bmatrix}\vec u & \vec v & \vec w \end{bmatrix} \quad \text{ and not like } \quad \begin{bmatrix} \vec u \\ \vec v \\ \vec w \end{bmatrix}\]

is so that the equation $\textbf{A} \vec x$ is written in an order analogous to $f(x)$ from vanilla function notation.

Remark: We could order the matrix along the rows if we started using row vectors as well. Then the order would be $\vec x \textbf{A}$. But this looks a little weirder, right?