Thread: Linear transformation problem

A linear transformation is given: such as:



Find and
I know that first you must prove that (1,1,1),(0,1,0) and(1,0,2) are linearly independent which is not a problem my problem is finding
Thanks in advance.

Originally Posted by wicked

A linear transformation is given: such as:



Find and
I know that first you must prove that (1,1,1),(0,1,0) and(1,0,2) are linearly independent which is not a problem my problem is finding
Thanks in advance.

Since (1,1,1), (0,1,0), and (1,0,2) are linearly independent (since you say "not a problem" I presume you have already proved that) and there are three vectors, they form a basis for . In particular, that means that any vector (x,y,z)= a(1,1,1)+ b(0,1,0)+ c(1,0,2) where a, b, and c are numerical functions of x, y, and z. After you have found those numbers, T(x,y,z)= aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1).

Originally Posted by HallsofIvy

(x,y,z)= a(1,1,1)+ b(0,1,0)+ c(1,0,2) where a, b, and c are numerical functions of x, y, and z. After you have found those numbers, T(x,y,z)= aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1).

Okay,so this is what I get




aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1) =
(2x-z)(1,1,1)+(y+z-2x)(0,1,0)+(z-x)(1,0,1)=(2x-z,2x-z,2x-z)+ (0,y+z-2x,0)+ (z-x,0,z-x)=(x,y,x) right?

So the matrix is :

( ) =

1 0 0
0 1 0 (this is the )
1 0 0
Is it correct?