Let $B$ be a basis of the vector space $V$. Show that a subset {v$_1$, . . . , v$_p$} in V is linearly independent if and only if the set of coordinate vectors {[v$_1]_B$, . . . , [v$_p]_B$} is linearly independent in $\mathbb{R}$$^{n}$.
This is for a first course in Linear Algebra, so it's not expected (or supposed) to be the most intricate or drawn-out.
Thoughts: Is it safe to assume that there is a coordinate mapping x ↦ [x]$_B$ (since it does not explicitly state this) ? Then, I could use the fact that the coordinate mapping is one-to-one and linear to write out a couple of equations that have the same solutions.
What do you think?
Yes it is safe to assume this. This mapping is called the standard representation of $\{v_1,...,v_p\}$ with respect to B. It's an isomorphism so linear indepenence will be preserved.
However, it's simpler to just write out what it means for a subset of $V$ to be linearly independent and carry through