Need help with graduate level work. I've been out of undergrad for almost 20 years. The problem says: Given $B$, a subset of a linear space $L$, $q$ element of $L$ and $p$ a point of $B$ such that when $q$ is a linear combination of points in $B$, the coefficient of $p$ is not $0$. Let $B'$ be the set obtained by replacing $p$ with $q$. Then $B'$ also spans $L$.
Is this just asking me to show that $p$ allows me to form a different linear combination which will also spans $L$ since $p$ is a part of the subset which spans $L$?
If so, How do I approach proving this?
(I think the omitted assumption is that $B$ spans $L$.) Starting with the given assumption $q = cp+\sum c_i b_i$ (for some scalars $c_i$ and some $b_i\in B$), write $$p=c^{-1}q-\sum c^{-1}c_i b_i \tag{1}$$ Using (1), you can eliminate $p$ from any linear combination in which it appears. Thus, anything that can be written as a linear combination of elements of $B$ can also be written as a linear combination of elements of $B'$.