December 8th, 2016, 09:21 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,757 Thanks: 900 
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 

Tags 
vector 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
vector as product of matrices, exp function of vector  whitegreen  Linear Algebra  1  June 9th, 2015 07:11 AM 
vector product of a vector and a scalar?  71GA  Algebra  1  June 3rd, 2012 02:20 PM 
Determining a 3D Vector B after a rotation of 3D Vector A  babarorhum  Algebra  0  October 20th, 2011 04:53 PM 
Vector Calculus Divergence of a Vector Field  MasterOfDisaster  Calculus  2  September 26th, 2011 10:17 AM 
Spherical Vector to Cartesian Vector  tsa256  Algebra  3  August 20th, 2010 08:33 AM 