- #1

- 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, Im 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.