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December 8th, 2016, 08:46 PM   #1
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Vector

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Stuck on this question, no idea what to do or how to solve it
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December 8th, 2016, 09:21 PM   #2
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let

$V=(v_1,~v_2,~v_3)$ be a $3 \times 3$ matrix whose columns are your 3 linearly independent $v$'s

let

$F$ be a $3 \times 3$ matrix representing your invertible linear transformation $f$

finally let

$B = (f(v_1),~f(v_2),~f(v_3)) = (Fv_1,~Fv_2,~Fv_3)=FV$

i.e. $B$ is the matrix whose columns are the $v$'s transformed by $F$

Now suppose the columns of $B$ are not linearly independent.

Then $\exists x \ni Bx =0$ so

$FVx=0$

But $F$ is invertible so multiply both sides by $F^{-1}$

$F^{-1}F V x = 0$

$Vx = 0$

and thus the column vectors of $V$ are linearly dependent.

but we are given they are linearly independent and thus we have a contradiction and it must be that the columns of $B$ are linearly independent.

I'll let you think about (b), it's important
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