Constructing a Dual Basis for V to Prove the Direct Sum of Dual Space

[yifli]

Homework Statement

show that if $$V=M \oplus N$$, then $$V^*=M^o+N^o$$

2. The attempt at a solution
So I need to prove for any $$f \in V*$$, $$f(\epsilon)=(g+h)(\epsilon)$$, where $$g\in M^o$$ and $$h\in N^o$$.

$$(g+h)(\epsilon)=g(\epsilon)+h(\epsilon)=g(\alpha+\beta)+h(\alpha+\beta)=g(\beta)+h(\alpha)$$, where$$\alpha \in M$$ and $$\beta \in N$$.

I'm stuck here, how to proceed?

Thanks

 

[Hurkyl]

Can you be more explicit in how you use symbols and what you're trying to do? You look like you're a little confused.

(P.S. are there typos in what you wrote? What is $M^o$? Did you really mean + instead of $\oplus$?)


This appears (to me) to be one of those problems where if you are clear and precise, everything is obvious -- so the only real obstacle is actually being clear and precise about what you're doing.

 

[yifli]

Homework Statement

show that if $$V=M \oplus N$$, then $$V^*=M^o+N^o$$

2. The attempt at a solution
So I need to prove for any $$f \in V*$$, $$f(\epsilon)=(g+h)(\epsilon)$$, where $$g\in M^o$$ and $$h\in N^o$$.

$$(g+h)(\epsilon)=g(\epsilon)+h(\epsilon)=g(\alpha+\beta)+h(\alpha+\beta)=g(\beta)+h(\alpha)$$, where$$\alpha \in M$$ and $$\beta \in N$$.

I'm stuck here, how to proceed?

Thanks

Sorry for the confusion. $$V$$ is a vector space and $$v^*$$ is the dual space.
M and N are the subspaces of V, and $$M^o$$ and $$N^o$$ are the annihilators. There was a type: $$V^*=M^o\oplus N^o$$, meaning direct sum

 

[wisvuze]

first, construct a dual basis for V ( the cannonical basis for V* ) -- that should get you far. Remember that the dual basis elements kill everything except for particular basis elements ( defined by a set { alpha_i } such that if { v_i } is a basis for V, alpha_i ( v_j ) = 1 when i = j and 0 otherwise ). Then remember that V is the DIRECT SUM of M and N, so that you know all about your basis for V.