Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 6

To be submitted by March 9, 12:00

1. Let V be a finite dimensional vector space and let W ⊆ V be a subspace. Prove that dim(V) = dim(W) + dim(V/W).

2. Define functionals ϕ_{1}, …, ϕ_{n} on ℝ[t]_{n} by

ϕ_{i}(f) = f(i).

Prove that {ϕ_{1}, …, ϕ_{n}} is a basis for (ℝ[t]_{n})^{⋆}. What basis {s_{1}, …, s_{n}} for ℝ[t]_{n} has the property that {ϕ_{1}, …, ϕ_{n}} is its dual basis?

3. Let U_{1}, U_{2} be subspaces of a vector space V. Prove that^{1} (U_{1} + U_{2})^{⊥} = U_{1}^{⊥} ∩ U_{2}^{⊥}, (U_{1} ∩ U_{2})^{⊥} = U_{1}^{⊥} + U_{2}^{⊥},

4. Let T : V ⟶ W be a linear transformation with kernel U. Show that Im(T^{⋆}) = U^{⊥}. Use that to conclude the following facts:^{2}

(1) If T is injective, T^{⋆} is surjective.

(2) If T is surjective, T^{⋆} is injective. (I suggest proving this fact directly from the definitions and not as a consequence of Im(T^{⋆}) = U^{⊥}).

(3) dim(Im(T)) = dim(Im(T^{⋆})).

(4) For every m × n matrix A we have rank_{r}(A) = rank_{c}(A).

1In your text book the notation U^{∘} is used instead of U^{⊥}.

2Try and prove the facts even if you can not prove that Im(T^{⋆}) = U^{⊥}.