# Homework Help: Prove dual space has the direct sum decomposition

1. Apr 16, 2012

### jpcjr

I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.

Any direction would be greatly appreciated!

Mike

1. The problem statement, all variables and given/known data

Let V be a vector space,

Let W1, ..., Wk be subspaces of V, and,

Let Vj = W1 + ... + Wj-1 + Wj+1 + ... + Wk.

Suppose that V = W1 $\oplus$ ... $\oplus$ Wk.

Prove that the dual space V* has the direct-sum decomposition V* = Vo1 $\oplus$ ... $\oplus$ Vok.

2. Relevant equations

See above.

3. The attempt at a solution

Again, I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 16, 2012

### micromass

What is the ${}^\circ$ in $V_j^\circ$?

3. Apr 16, 2012

### jpcjr

What is the " o " in Voj?

Definition.
If V is a vector space over the field F and S is a subset of V, the annihilator of S is the set So of linear functionals f on V such that f(α) = 0 for every α in S.
.
.
.

4. Apr 16, 2012

### micromass

OK. You must prove

$$V^*=V_1^\circ \oplus ... \oplus V_n^\circ$$

What does that mean?? What is the definition of a direct sum? What is it you need to check?

5. May 8, 2012

### jpcjr

Thank you!

By the skin of my teeth, some help from you, and the grace of God, I received the best grade I could have expected in Linear Algebra.

Thanks, again!

Joe

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