The proof in question can be found on page 185-186 of this textbook https://joshua.smcvt.edu/linearalgebra/book.pdf (it is too long to type everything out). The question asks if this proof is valid when the dimension n = 0, and the answer simply states that "Yes, because a zero-dimensional space is a trivial space", which sounds completely circular to me, since it basically just states the definition of a trivial space! The most confusing part, to me, is at the bottom of page 185, where it states
$\vec{v}$ = $v_1 {\beta_1}$ + ... + $v_n {\beta_n}$ $\xrightarrow{Rep_B}$ $\begin{pmatrix}v_1 \\ ... \\v_n\end{pmatrix}$
Here, as I underestand it, the basis for the trivial space is the empty set. But how would that convention go in the above equation? Would it be $\vec{v}$ = {} $\xrightarrow{Rep_B}$ ( ) ? Does such a mathematical statement even exists when dealing with empty set as your basis? Thanks in advance!
