Question

Answer for all the parts will be appreciated, since they are parts of the same question

「-4-30 blem 1. (20 pts) Let A- 0 14 Problem I. (20 pts) Let A 0 3and b 1 (a) (10 pts) Determine if b is in the span of the columns of A. (b) (5 pts) Determine if the columns of A are linearly independent. Compute the null space of A (c) (5 pts) What is the rank of A? What is the Nullity of A?

Answer 1

(a). Let M = [A|b] =

-4	-3	0	1
0	-1	4	1
1	0	3	1
5	4	6	1

The RREF of M is

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

This implies that b cannot be expressed as a linear combination of the columns of A. Hence b is not in the span of the columns of A.

(b). It may be observed from the RREF of that the RREF of A is

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

This implies that the columns of A are linearly independent.

( c). Since the RREF of A has 3 non-zero rows, hence the rank of A is 3. Further, as per the dimension theorem, the nullity of A = number of columns of A- rank of A = 3-3 = 0.