Question

Exercise 4.10.27 Here are some vectors in R4 21 r11 3 4 04 Thse vectors can't possibly be linearly independent. Tell why. Next obtain a linearly independent subset of these vectors which has the same span as these vectors. In other words, find a basis for the span of these vectors.

Answer 1

these vectors can't possibly be linearly independent....because there are 5 columns and 4 rows.....in other words there are 5 variables and 4 equation......for linearly independent ....there are must be 5 equation or 5 rows....

now construct a matrix form

1	1	1	2	1
3	4	0	-1	4
-1	-1	-1	-2	0
1	1	1	2	1

Add (-3 * row1) to row2

1	1	1	2	1
0	1	-3	-7	1
-1	-1	-1	-2	0
1	1	1	2	1

Add (1 * row1) to row3

1	1	1	2	1
0	1	-3	-7	1
0	0	0	0	1
1	1	1	2	1

Add (-1 * row1) to row4

1	1	1	2	1
0	1	-3	-7	1
0	0	0	0	1
0	0	0	0	0

Add (-1 * row3) to row2

1	1	1	2	1
0	1	-3	-7	0
0	0	0	0	1
0	0	0	0	0

Add (-1 * row3) to row1

1	1	1	2	0
0	1	-3	-7	0
0	0	0	0	1
0	0	0	0	0

Add (-1 * row2) to row1

1	0	4	9	0
0	1	-3	-7	0
0	0	0	0	1
0	0	0	0	0

there are 3 pivot entry at first, second , and fifth column.........so span of vectors are

1

3

-1

1

1

4

-1

1

1

4

0

1