Orthonormal set

1. Mar 30, 2007

EvLer

I just need a hint.
Problem:
find an orthonormal set q1, q2, q3 for which q1, q2 span the column space of A, where
A =
[1 1]
[2 -1]
[-2 4]

of course I should apply the Gram-Schmidt method, but the problem is that the column vectors are not independent and Gram-Schmidt starts with independent vectors. I can arrive at independent columns by Gaussian elimination that would span the subspace of the column space, so is that what I should use for finding orthonormal qs?
Thanks.

2. Mar 30, 2007

Hurkyl

Staff Emeritus
The two columns look independent to me.

3. Mar 30, 2007

EvLer

but if I take inner product i don't end up with 0...

4. Mar 30, 2007

Hurkyl

Staff Emeritus
Then they aren't orthogonal.

5. Mar 31, 2007

HallsofIvy

Staff Emeritus
Of course they aren't! You are asked to find an orthonormal basis. You would hardly expect them to give you vectors that are already orthogonal!

The two vectors you are given definitely are independent- the only way two vectors can be dependent is if one is a multiple of the other and that clearly is not the case here.

Apply "Gram-Schmidt" to them!

6. Mar 31, 2007

D H

Staff Emeritus
Since A is 2x3, you can also use a trick that works in R3 only. The vector cross product is very useful for constructing an orthonormal set.

7. Mar 31, 2007

EvLer

oh, ok, I know where I got confused: othogonal is always independent, independent may not be orthogonal... thanks