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If $\{v_{1},v_{2},\cdots,v_{n}\}$ is $n$ linearly independent vectors in $\mathbb{R}^{n}$, what would be necessary and sufficient condition of $A$ ($n\times n$ matrix) $A$ so that the vectors $Av_{1}$, $Av_{2}$, $\cdots$, $Av_{n}$ are linearly independent?

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1 Answer 1

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Hint:

The vectors $Av_1, \ldots ,Av_n$ are linearly independent if, and only if, $\det \left([Av_1 | \ldots |Av_n]_{n\times n}\right)\neq 0$.

Now note that $[Av_1 | \ldots |Av_n]_{n\times n}=A[v_1|\ldots |v_n]_{n\times n}$.

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why is ther vertical lines in the matrix, before and after the dots? –  Mark Oct 9 '13 at 16:49
    
@Mark It's a notation which suggests that I'm 'writing the matrix by columns'. The $i^{\text{th}}$ column of the matrix $[Av_1 | \ldots |Av_n]_{n\times n}$ is $Av_i$. Is that clear now? –  Git Gud Oct 9 '13 at 16:52
    
yes so you gave me a property of matrix with column vectors that are linearly independent. So I need to find the condition of A that makes the matrix product with nonzero determinant correct? –  Mark Oct 9 '13 at 17:03
    
So, the matrix product $\left([Av_1 | \ldots |Av_n]_{n\times n}\right)$ has to be invertible here –  Mark Oct 9 '13 at 17:05
    
@Mark Yes, but that's not what I'm hinting at. Think about the last line in my answer. –  Git Gud Oct 9 '13 at 17:09

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