a 1 3 3 1 5 1 2 7 4 1 4 1 b 1 4 1 3 2 8 3 9 1 4 0 0 2 8 2 6S

a) {(1, 3, 3), (1, 5, 1), (2, 7, 4), (1, 4, 1)}

b) {(1, 4, 1, 3), (2, 8, 3, 9), (1, 4, 0, 0), (2, 8, 2, 6)}

Solution

a) Let A =

1

1

2

1

3

5

7

4

3

-1

4

1

We will reduce A to its RREF as under:

Then the RREF of A is

1

0

3/2

½

0

1

½

½

0

0

0

0

Apparently, only the first two vectors are linearly independent and the 3^rd and the 4^th vectors are linear combination of the first two vectors. Therefore, a basis for the given set is {(1, 3, 3)^T, (1, 5, 1)^T}.

b) Let A =

1

2

1

2

4

8

4

8

1

3

0

2

3

9

0

6

We will reduce A to its RREF as under:

Then the RREF of A is

1

0

3

2

0

1

-1

0

0

0

0

0

0

0

0

0

Apparently, only the first two vectors are linearly independent and the 3^rd and the 4^th vectors are linear combination of the first two vectors. Therefore, a basis for the given set is {(1, 4, 1, 3)^T, (2, 8, 3, 9)^T}   

1	1	2	1
3	5	7	4
3	-1	4	1