- #1

Cade

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## Homework Statement

Describe every solution to Ax=0 where A is:

1 2 2 4 6

1 2 3 6 9

0 0 1 2 3

## Homework Equations

I'm not sure.

## The Attempt at a Solution

I found the echelon form of A to be:

1 2 2 4 6

0 0 1 2 3

0 0 0 0 0

Pivot variables: x1, x3

Free variables: x2, x4, x5

Finding special solutions:

x1 + 2x2 + 2x3 + 4x4 + 6x5 = 0

x3 + 2x4 + 3x5 = 0

x2 = 1, x4 = 0, x5 = 0 -> x3 = 0, x1 = -2 (-2, 1, 0, 0, 0)

x2 = 0, x4 = 1, x5 = 0 -> x3 = -2, x1 = 0 (0, 0,-2, 1, 0)

x2 = 0, x4 = 0, x5 = 1 -> x3 = -3, x1 = 0 (0, 0,-3, 0, 1)

If this is the nullspace, how do I describe every solution? I thought of writing as a linear combination of those three, but that's apparently the wrong answer, my instructor wants a single vector like (*, x2, *, x4, x5). where * is a multiple of one of the pivot variables.

Edit: Looking at the special solutions, I think (-2x2, x2, -2x4 -3x5, x4, x5) is the right answer, but I'm not sure.

I also have a second problem: With pivot variables x1, x3 and free variables x2, x4 and x5 with the solution with x3=1 (1,-1,1), is the general solution (x3,-x3,x3)? My textbook says it should be (2x3,-x3,x3).