(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the matrix A:

1 4 5 0 9

3 -2 1 0 -1

-1 0 -1 0 -1

2 3 5 1 8

(Sorry I don't know how to do TeX matrices on this site)

Find a basis for the row, column, and null space.

2. Relevant equations

3. The attempt at a solution

I reduced to row echelon form, which got me:

1 0 1 0 1

0 1 1 0 2

0 0 0 1 0

0 0 0 0 0

Row space:

I took all non-zero rows to be the vectors for the row space

Column space:

I found the columns from the ref version that were linearly independent.

v1 = [ 1 0 0 0 ]

v2 = [ 0 1 0 0 ]

v3 = [ 1 1 0 0 ]

v4 = [ 0 0 1 0 ]

v5 = [ 1 2 0 0 ]

v1+2*v2 = v5, so those 3 are dependent.

v1+v2 = v3, so those 3 are dependent.

Only v4 is independent since no combinations of the others are equal to it, and no combination of it is equal to any of the others, so I just take v4 as the column space? A bit confused here but that is my understanding.

Null space:

in row echelon

x1 = -x3 -x5

x2 = -x3 -2x5

x4 = 0

So x3 and x5 are the 'free variables' but I'm not sure where to go from here.

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# Homework Help: Row, column, null spaces.

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