For the given matrix A and vector b select the true statemen

For the given matrix A and vector b, select the true statements. A = [1 5 0 1 5 1 5 1 0 4 2 1] b = [3 2 1] The dimension of CoLA is 2. NuLA is a subspace of R^3. b is in NulA. CoLA is all of R^3. The dimension of NuLA is 1. b is in CoLA.

Solution

Let B = [ A|b] =

1

5

0

1

3

5

1

5

1

2

0

4

2

1

1

The RREf of B is

1

0

0

3/68

49/68

0

1

0

13/68

31/68

0

0

1

2/17

-7/17

Then,

1.dim( Col(A)) = 3;

2. Null(A) being the set of solutions to the equation AX = 0, is a subspace of R⁴;

3.b is a 3-vector and Null(A) contains 4-vectors so that b cannot be in Null(A);

4.Col(A) is all of R³ as its columns are linear combinations of (1,0,0)^T, (0,1,0)^T and (0,0,1)^T;

5.If X = (x,y,z,w)^T, then Null(A) is the set of solutions to the equations x +3w/68 = 0, y +13w/68 = 0 and z +2w/17 = 0 so that X = (-3w/68,-13w/68, -2w/17,w)^T = w/68( -3,-13,8,68)^T. Thus Null(A) = span{( -3,-13,8,68)^T } so that dimension of Null(A) is 1;

6. b = (49/68)(1,5,0 )^T + (31/68)(5,1,4 )^T-(7/17)(0,5,2 )^T so that b is in Col(A).

The statements numbers 4,5, and 6 are true.

1	5	0	1	3
5	1	5	1	2
0	4	2	1	1