Question

Find bases for the four fundamental subspaces of the matrix A

[ 1 0 0 ]

0 1 1  

1 1 1

[ 1 8 8 ]

find N(A) basis ______________

N(A^T) = __________

R(A) basis = ___________

R(A^T) = ______________

Answer 1

for matrix A

convert into Reduced Row Eschelon Form...

1	0	0
0	1	1
1	1	1
1	8	8

Add (-1 * row1) to row3

1	0	0
0	1	1
0	1	1
1	8	8

Add (-1 * row1) to row4

1	0	0
0	1	1
0	1	1
0	8	8

Add (-1 * row2) to row3

1	0	0
0	1	1
0	0	0
0	8	8

Add (-8 * row2) to row4

1	0	0
0	1	1
0	0	0
0	0	0

take transpose of matrix

so =

1	0	1	1
0	1	1	8
0	1	1	8

convert into Reduced Row Eschelon Form...

Add (-1 * row2) to row3

1	0	1	1
0	1	1	8
0	0	0	0