Finding Orthonormal Set q1, q2, q3 for Column Space of A

[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.

 

[Hurkyl]

The two columns look independent to me.

 

[EvLer]

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

 

[Hurkyl]

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

Then they aren't orthogonal.

 

[HallsofIvy]

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

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!

 

[D H]

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.

 

[EvLer]

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