I need help understanding the above proof. Why did we chose the vectors $\{\mathbf{w}_1, \mathbf{w}_2, \dots , \mathbf{w}_n\}$ or define them in this way?
How does the nonsingularity of $Q$ imply that $\beta$ is an ordered basis for $V$?
Obviously, there are simple answers to these, which I can't see. Please try to hint at the thought process that is being employed in this proof.
Hint:
For 1. Try to construct a new $n$ linearly independent basis vectors from the original $n$ linearly independent basis vectors so that Theorem 4.14 (3) can be used in the end. But just writing down $w_1,\cdots,w_n$ isn't sufficient for them to be linearly independent.
For 2. $Q$ non-singular implies that $Q$ is full rank matrix and thus the above constructed basis vectors are linearly independent. (Do Row Echelon on Q in $W=QV$ if not convinced right away.)