For each of the following sets of vectors decide whether the

For each of the following sets of vectors, decide whether the set is linearly dependent or linearly independent. You should completely justify your answers. For those sets that are linearly dependent, write one of the vectors as a linear combination of the others. {(2, -2, 4), (3, -5, 4), (0, 1, 1)} {(3, 8, 7, -3), (1, 5, 3, -1), (2, -1, 2, 6), (1, 4, 0, 3)} {(2, -1, 4, 1), (4, 2, 3, 2), (2, 7, -6, 1)}

Solution

(a) Let A =

2

3

0

-2

-5

1

4

4

1

The RREF of A is

1

0

3/4

0

1

-1/2

0

0

0

Hence (0,1,1)^T = 3/4(2,-2,0)^T-1/2(3,-5,4)^T.The given set is linearly dependent.

(b) Let B =

3

1

2

1

8

5

-1

4

7

3

2

0

-3

-1

6

3

The RREF of B is I_4. Therefore, the given set is linearly independent.

( c ) Let C =

2

4

2

-1

2

7

4

3

-6

1

2

1

    The RREF of C is

1

0

-3

0

1

2

0

0

0

0

0

0

Therefore, (2,7,-6,1)^T = -3(2,-1,4,1)^T+2(4,2,3,2)^T. The given set is linearly dependent.

2	3	0
-2	-5	1
4	4	1