Unit Vectors of Cartesian 3-Space form Basis

Theorem
Consider the Cartesian $3$-space $C$.

Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.

Then $\set {\mathbf i, \mathbf j, \mathbf k}$ forms a basis for $C$.