Question

A=

1	2	4	0	1
-2	-1	0	1	2
0	3	8	1	4

. Let W denote the row space for A.

(a) Find an orthonormal basis for W and for W⊥.

(b) Compute projW⊥(1 1 1 1 1 ).

Answer 1

a) Given is the matrix

The space is row space. We perform Gram Schmidt on the row vectors of the above matrix. Let

Let

Then is an orthonormal basis of .

To find orthonormal basis of we first find a basis of it. Note that is the null space of the above matrix. Therefore, we need to perform elementary row operations. We get

Thus, a typical null vector satisfies

Thus, basis for is

To orthonormalize this, we do Gram-Schmidt: Let

Let

and

Then is an orthonormal basis for .

b) We have