Let $V$ be a nonzero finite-dimensional vector space over a field $F$ and $\beta=\{v_1,…,v_n\}$ be an ordered basis for $V$. Let $Q$ be an invertible $n\times n$ matrix with entries from $F$.
Now, define $v_j'=\sum_{i=1}^n Q_{ij}v_i, \forall 1≦j≦n$, and let $\beta'=\{v_1',…,v_n'\}$.
So far, i have proved that $\beta'$ is linearly independent, but how do i prove that it is a basis for $V$? That is, how do i know there does not exist a pair $(i,j)$ such that $i≠j \bigwedge v_i'=v_j'$?
Best Answer
Since $ V $ is a vector space of dimension $ n $. I know that any linearly indepedent set containing $ n $ vector basis of $ V $.