December 8th, 2016, 09:21 PM  #2 
Senior Member Joined: Sep 2015 From: CA Posts: 1,237 Thanks: 637 
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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