(a) | Prove the following implication. If $w \notin {\rm span}\, \mathcal A$, then $ w + \mathcal A$ is a linearly independent set. |
(b) | Is the converse of the implication in (a) true? Justify your claim. |
(c) | Let $\alpha_1,\cdots,\alpha_n \in \mathbb F$, let $v_1,\ldots, v_n$ be distinct vectors in $\mathcal A$ and let $w = \alpha_1 v_1 + \cdots + \alpha_n v_n$. Find a necessary and sufficient condition (in terms of $\alpha_1, \ldots, \alpha_n$) for the linear independence of the vectors $v_1 + w, \ldots, v_n + w$. |