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

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Homework Help Overview

The problem involves finding an orthonormal set of vectors q1, q2, q3 that span the column space of a given matrix A, which is a 2x3 matrix. The original poster expresses uncertainty regarding the application of the Gram-Schmidt method due to concerns about the independence of the column vectors.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster considers using Gaussian elimination to obtain independent columns from the matrix A before applying Gram-Schmidt. Some participants question the independence of the columns, while others assert their independence and suggest applying Gram-Schmidt directly.

Discussion Status

The discussion is active, with participants exploring different interpretations of vector independence and orthogonality. Some guidance has been offered regarding the application of Gram-Schmidt, and alternative methods such as the vector cross product have been mentioned for constructing an orthonormal set.

Contextual Notes

There is a noted confusion between the concepts of orthogonality and independence, with participants clarifying that while orthogonal vectors are independent, independent vectors are not necessarily orthogonal. The matrix A's dimensions and the implications for the column space are also relevant to the discussion.

EvLer
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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.
 
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The two columns look independent to me.
 
but if I take inner product i don't end up with 0...
 
EvLer said:
but if I take inner product i don't end up with 0...
Then they aren't orthogonal.
 
EvLer said:
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!
 
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.
 
oh, ok, I know where I got confused: othogonal is always independent, independent may not be orthogonal... thanks
 

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