Let A 2 6 10 8 24 40 7 24 41 4 7 8 3 12 27 u 1 2 2 1 0 and

Let A = [2 6 10 -8 -24 -40 7 24 41 -4 -7 -8 3 12 27], u = [1 -2 -2 1 0] and b = [-4 5 4]. Is u in the null space of A? Explain Is b in the column space of A? (If it is, express b as a specific linear combination of the columns of A (the coefficients should be numbers). If it isn\'t, explain why not. Find the null space of A. Find the null space of A. Find the column space of A. Find rank A.

Solution

Let B = [A|b] =

2

-8

7

-4

3

-4

6

-24

-7

12

5

10

-40

41

-8

27

4

We will reduce B to its RREF as under:

Multiply the 1st row by ½ ; Add -6 times the 1st row to the 2nd row

Add -10 times the 1st row to the 3rd row; Multiply the 2nd row by 1/3

Add -6 times the 2nd row to the 3rd row; Multiply the 3rd row by ½

Add -5/3 times the 3rd row to the 2nd row ; Add 2 times the 3rd row to the 1st row

Add -7/2 times the 2nd row to the 1st row

Then the RREF of B is

1

-4

0

0

-43/2

-61

1

0

1

0

-4

14

0

0

0

1

3

-5

Apparently,

4

-43/2

1

1

0

-2

0

4

-2

0

-3

1

0

1

0

The RREF of C is

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

This means that u is not in Null(A).

2	-8	7	-4	3	-4
6	-24	24	-7	12	5
10	-40	41	-8	27	4