Question

7. Determine if the following vectors in R' are linearly independent. a (2,0,-4,6)b (1,5,0,-3) c (1,-15,-8,21)

Answer 1

construct a matrix form and convert into RREF

2	1	1
0	5	-15
-4	0	-8
6	-3	21

Divide row1 by 2

1	1/2	1/2
0	5	-15
-4	0	-8
6	-3	21

Add (4 * row1) to row3

1	1/2	1/2
0	5	-15
0	2	-6
6	-3	21

Add (-6 * row1) to row4

1	1/2	1/2
0	5	-15
0	2	-6
0	-6	18

Divide row2 by 5

1	1/2	1/2
0	1	-3
0	2	-6
0	-6	18

Add (-2 * row2) to row3

1	1/2	1/2
0	1	-3
0	0	0
0	-6	18

Add (6 * row2) to row4

1	1/2	1/2
0	1	-3
0	0	0
0	0	0

Add (-1/2 * row2) to row1

1	0	2
0	1	-3
0	0	0
0	0	0

there are 3 column but only 2 pivot entry. for the linearly independent there are must be 3 pivot entry

hence vectors are NOT linearly independent