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**Mod note: Moved from Precalc section**

1. Homework Statement

1. Homework Statement

Given l : IR3 → IR3 , l(x1, x2, x3) = (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3, x1 + x2 + x3), find Ker(l), Im(l), their bases and dimensions.

My language in explaining my steps is a little sloppy, but I'm trying to understand the process and put it in terms that I understand.

## Homework Equations

ker(l) Ax=0

## The Attempt at a Solution

Step 1: Find the matrix associated to this transformation using the standard basis.

l(1,0,0) = (1,4,1)

l(0,1,0) = (2,5,1)

l(0,0,1) = (3,6,1)

Im(l) =

[1] [4] [1]

[2] , [5] , [1]

[3] [6] [1]

To find Ker(l) = Ax=0. Put Im(l) into an augmented matrix and set it =0

[1 4 1 |0]

[2 5 1 |0]

[3 6 1 |0]

Reduced row echelon form =

[1 0 -1/3 |0]

[0 1 1/3 |0]

[0 0 0 |0]

X1= X1 [1] X2 [0] X3 [ -1/3]

X2= X1 [0] + X2 [1] + X3 [ 1/3 ]

X3= X1 [0] X2 [0] X3 [ 0 ]

ker(l) =

[1] [0] [-1/3]

[0] , [1] , [1/3]

[0] [0] [0]

Basis(l) = minimum number of vectors that span the subspace

[1] [0]

[0] , [1]

[0] [0]

Dimension = number of vectors that span the subspace = 2

I hope I've done this right.

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