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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 $i≠j \bigwedge v_i'=v_j'$?

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    $\begingroup$ I got it just after i posted this.. Please close this question.. Sorry $\endgroup$ – Jj- Mar 4 '13 at 12:54
  • $\begingroup$ As owner you can in fact delete the question, which is better. $\endgroup$ – Marc van Leeuwen Mar 4 '13 at 13:00
  • $\begingroup$ Even better. Write up the answer to your question. $\endgroup$ – Thomas Mar 4 '13 at 14:01
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Since $ V $ is a vector space of dimension $ n $. I know that any linearly indepedent set containing $ n $ vector basis of $ V $.

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