This is a subject I struggled with the first time I took it. Ironically, this was the engineering version of it. It wasn't until I took the rigorous, axiomatic version that everything clicked.

Suppose we have a subspace \(\mathbb{S}\) of \(\mathbb{R}^n\) whose basis consists of \(k\) vectors \(\vec{v}_1,\vec{v}_2, \ldots , \vec{v}_k\). \[ \mathbb{S ...