Matrix Product (Conventional)/Examples/Change of Axes

Example of (Conventional) Matrix Product
Consider theCartesian coordinate system:
 * $C := \tuple {x, y, z}$

Let $\mathbf A$ denote the square matrix:
 * $\mathbf A = \begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$

Then $\mathbf A$ has the effect of exchanging the $x$ and $y$ axes of $C$.

Proof
Let $\mathbf x := \tuple {x_1, y_1, z_1}$ be a point in $C$.

We have: