Abstract Linear Algebra: Dual Basis

Homework Statement

Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0.

N/A

The Attempt at a Solution

Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be the dual basis in C3'. Let N be the subspace of V' spanned by ym+1, ..., yn.
Let's assume that y is any element in N.
1) y is in V'
2) y is a linear combination of the basis vectors y1, ..., yn
=> y = $$\Sigma$$j=1n njyj
Since by assumption y is in N we have for every i=1,...,m
[xi,y] =0

P.S: I am new to the abstract linear algebra world.

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Dick
Homework Helper
Apparently you mean a linear functional on C^3. What is C? Is it the complex numbers? And I think you are taking this 'abstract' thing a little far. You just want a linear functional y such that y((1,1,1))=y((1,1,-1))=0. Try thinking of it as a 'not abstract' problem. You want to write down a concrete linear functional that maps x1 and x2 to zero.

Ok, so let x=($$\xi$$1,$$\xi$$2,$$\xi$$3) (where $$\xi$$1=$$\xi$$2) and let y be the functional such that y = $$\xi$$1+$$\xi$$2+0*$$\xi$$3
So for xi (i from 0 to n) y(xi) would equal 0.

Last edited:
Dick
Ok, so let x=($$\xi$$1,$$\xi$$2,$$\xi$$3) (where $$\xi$$1=$$\xi$$2) and let y be the functional such that y = $$\xi$$1+$$\xi$$2+0*$$\xi$$3