Projections a What vector bL in L span 2 1 1 1 is closest t

Projections (a) What vector b_L in L = span {(2, -1, 1, 1)} is closest to b = (1, 2, 3, 4)? What vector b_P in P = span {(1, 2, -2), (1, 1, 0)} is closest to b = (2, 1, 1)?

Solution

(a) Let A =

2

1

-1

2

1

3

1

4

We will reduce A to its RREF as under:

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

Add -1 times the 1st row to the 3rd row ; Add -1 times the 1st row to the 4th row

Multiply the 2nd row by 2/5 ; Add -5/2 times the 2nd row to the 3rd row

Add -7/2 times the 2nd row to the 4th row ; Add -1/2 times the 2nd row to the 1st row

Then the RREF of A is    

1

0

0

1

0

0

0

0

Apparently, no vector in L can be close to the vector b = ( 1,2,3,4)^T.

(b) Let B =

1

1

2

2

1

1

-2

0

1

We will reduce B to its RREF as under:

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

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

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

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

       Then the RREF of B is                

1

0

0

0

1

0

0

0

1

Apparently, no vector in P can be close to the vector b = (2,1,1)^T

2	1
-1	2
1	3
1	4