## Spanning and independence, regarding the number of dimensions

Note:

Let $W = [w_1, \cdots w_n]$ and $V = [v_1, \cdots, v_n]$ , then we have $W = VC^T$, and $V = W(C^T)^{-1}$, i.e. W also spans the vector space, and $w_{n+1}$ is dependent on the set $W$.

From this lemma, an important result can be proved:

The proof is easy enough, but I want to give it a visualization here: