- #1

- 7

- 0

## Homework Statement

Define a non-zero linear functional y on C^2 such that if x

_{1}=(1,1,1) and x

_{2}=(1,1,-1), then [x

_{1},y]=[x

_{2},y]=0.

## Homework Equations

N/A

## The Attempt at a Solution

Le X = {x1,x2,...,xn} be a basis in C

^{3}whose first m elements are in M (and form a basis in M). Let X' be the dual basis in C

^{3}'. Let N be the subspace of V' spanned by y

_{m+1}, ..., y

_{n}.

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 y

_{1}, ..., y

_{n}

=> y = [tex]\Sigma[/tex]

_{j=1}

^{n}n

_{j}y

_{j}

Since by assumption y is in N we have for every i=1,...,m

[x

_{i},y] =0

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