Question 29839: I have to solve this proof but dont understand it all.
Here it is: 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.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! 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
|
|
|
