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Simple, construct a basis of R2 with an inner product

  • Thread starter simpledude
  • Start date
21
0
1. Homework Statement
Construct an orthogonal basis of R2 equipped with the non-standard inner product defined for all X, Y belonging to R2, by

<X,Y> = X^T AY
with
A =
2 1
1 3

3. The Attempt at a Solution
So it seems pretty trivial, but I can't seem to get the answer. So my approach is
1) First get any basis that fits with this inner product
2) Use Gram-Scmidt to orthogonalize that basis

So I know I can start with the canonical basis of R2: (1,0) (0,1)
My problem is, how can I satisfy the inner product if I start from the canonical basis?

I have also attempted to solve the equations of two arbitrary vectors v1, v2
where v1=(x1, y2) and v2=(x2,y2) and given the matrix A we know:
x1x2 + y1y2 = 1
x1^2 + y1^2 = 2
x2^2 + y2^2 = 3
But if I plug in the canoical basis for v1, the solution fails. It only works when both points are non-zero, i.e. v1 = (1, 1)


Any help would be appreciated.

EDIT:
Actually now that I am looking at my solution, I obtained (via my method of substituting (1,1) for x1, y1 in my equations) the basis that works with this inner product as:
( 1 - [1+sqrt(5)]/2 , [1+sqrt(5)]/2 ) and (1, 1)

They seem to be linearly independent, did I do this right? (I know I still have to orthogonalize them)
 
Last edited:

Answers and Replies

21
0
Anyone?
 

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