Using the matrix A 1 2 1 1 1 2 2 1 2 4 0 6 and the column v

Using the matrix A =

1 2 1 1

1 2 2 -1

2 4 0 6

and the column vector b =

7

12

4

(a) State the rank of A. Is the column space of A a line, a plane, or all of R^3 ? Justify your answer.

(b) Find the general solution to the system Ax = b.

Solution

Let B = [ A| b] =

1

2

1

1

7

1

2

2

-1

12

2

4

0

6

4

We will reduce B to its RREF as under:

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

Add -2 times the 1st row to the 3rd row

Add 2 times the 2nd row to the 3rd row

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

Then the RREF of B is

1

2

0

3

2

0

0

1

-2

5

0

0

0

0

0

(a)   Thus, the rank of A is 2 and only the vectors (1,1,2)^T and (1,2,0)^T are linearly independent. The other two columns of A are linear combinations of these two columns. . The column space of A is a plane spanned by its 1^st and 3^rd columns.

(b)   If X = (x,y,z,w)^T , then the equation AX = b is equivalent to x +2y +3w = 2 and z-2w = 5. Let y = r and w = t. Then x = 2- 2r-3t and z = 5+2t so that X = (2- 2r-3t, r, 5+2t,t)^T = (2,0,5,0)^T + r (-2,1,0,0)^T +t (-3,0,2,1)^T