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dcramps
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Homework Statement
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.
Homework Equations
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.