Question 29839
 You have to know a linear transformation T:V-->V  is invertible(one-to-one)
 <--> Ker(T)(null space of T) = {0}
 Let v and w be independent column vectors in R^3, and let A be 
 an invertible 3X3 matrix.

Prove that the vectors Av and Aw are independent. 
 Proof: A: R^3-->R^3 invertible <--> N(A) = {0} <--> rank A = 3.
 a Av + b Aw = 0 for two reals a, b
 --> A(av+bw) = 0
 --> av + bw belongs to N(A)= {0}
 --> av+vw = 0
 --> a=b = 0 (since v, w are lindep.)
 Hence, Av & Aw are l.indep.

 This fact is true for any number of indep vectors.
 Also, it seems you have to work hard.

 Kenny