Explain why each orthogonal statement cannot be true Explain

Explain why each orthogonal statement cannot be true.

Explain why each statement cannot be true. 1. u = Span {[0, 0, 1], [1, 2, 0]} and w = Span {[1, 0, 0], [1, 0, 1]}, and there is a vector space v that is the orthogonal complement of u in w. 2. u = Span {[3, 2, 1], [5, 2, -3]} and w = Span {[1, 0, 0], [1, 0, 1], [0, 1, 1]} and the orthogonal complement v of u in w contains the vector [2, -3, 1].

Solution

1

1

0

2

0

0

1

-3

0

1

1

1

We will reduce A to its RREF as under:

Interchange the 2nd row and the 3rd row

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

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

Then, the RREF of A is

1

0

0

-2

0

1

0

4

0

0

1

-3

Therefore X = ( 2,-3,1) = -2(1,0,0)+4(1,0,1)-3(0,1,1) W.

If X = (2,-3,1) V = U in W, then we must have X.(0,0,1)= 0. However, X.(0,0,1)= 1 0. Also, X.(1,2,0) = 2-6 = -4 0.Thus, X = (2,-3,1) V = U. Therefore, the given statement is not true.

1	1	0	2
0	0	1	-3
0	1	1	1