Question

. Enlarge each of the following linearly independent subsets T of R5 to a basis B for R5 containing T :

(a) T {[1,3, 0, 1, 4],[2, 2, 1,3, 1]}

(b) T {[1, 1, 1, 1, 1],[0, 1, 1, 1, 1],[0, 0, 1, 1, 1]}

(c) T {[1, 0,1, 0, 0],[0, 1,1, 1, 0],[2, 3,8,1, 0]}

Answer 1

(a). Let A =

1	2
3	2
0	1
1	3
4	1

The RREF of A is

1	0
0	1
0	0
0	0
0	0

Hence, the set { [1,3,0,1,4],[ 2,2,1,3,1], [0,0,1,0,0], [0,0,0,1,0], [0,0,0,0,1]} includes T and is a basis for R⁵ .

(b). Let B =

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

The RREF of B is

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

Hence, the set { [1,1,1,1,1],[ 0,1,1,1,1, [0,0,1,1,1], [0,0,0,1,0], [0,0,0,0,1]} includes T and is a basis for R⁵ .

(c ). Let C =

1	0	2
0	1	3
1	1	8
0	1	1
0	0	0

The RREF of B is

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

Hence, the set {[1,0,1,0,0], [0,1,1,1,0], [2,3,1,,0], [0,0,0,1,0], [0,0,0,0,1]} includes T and is a basis for R⁵ .