- #1
says
- 594
- 12
Homework Statement
Given the linear transformation l : R2 → R2 , l(x, y) = (2x − 2y, −x + y), write the matrix associated to l with respect to the standard basis of R2 , find Kerl, I am l, its bases and dimensions. Find all vectors of R2 that are mapped to (4, −2).
Homework Equations
Ax=0 (Nullspace / Kernel)
The Attempt at a Solution
standard basis in R2 = (1,0) and (0,1)
L(1,0) = (2,-1)
L(0,1)= (-2,1)
so matrix A =
[2 -2]
[-1 1]
If we row reduce this matrix we see that there is a free variable, which means the basis of this matrix =
[2 ]
[-1]
Dimension = 2
Im(l) =
{ [ 2] , [-2] }
[-1] , [ 1]To get the kernel we make the matrix A =
[2 -2 | 0]
[-1 1 | 0]
Row reduction gets me to
[1 1/2 | 0]
[0 0 | 0]
ker(l) =
{ [ 1] , [1/2] }
[ 0] , [ 1 ]
[x] = X [1] - Y[1/2]
[y] = [0] [1]
Vectors mapped to (4,-2)
R2 → R2 , l(x, y) = (2x − 2y, −x + y)
[2(4) - (2)(-2), -(4)+(-2)]
= (12, -6)
I'm not sure if I've done it correctly, this is the first time I've tackled a problem like this. The fact that the ker(l) has a free variable makes me think that maybe I did something wrong -- I'm just reading up on linear transformations and linear independence / dependence now.