University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 12: Due November 30, 2007

1. Do problems 1, 2, 3, 4, 11, 12, 13, 14 on page 329-330.
   
   
2. Do problems 1, 2, 3, 4, 6, 7 on page 348.
   
   
3. Let V be a vector space with inner product ⟨,⟩ and orthonormal basis b₁, b₂, ..., b_n. Let v be a vector in V. Explain how to use ⟨,⟩ to write v as a linear combination of b₁, b₂, ..., b_n.
   
   
4. Let V be a vector space with inner product ⟨,⟩ and let W be a subspace of V. Define W^⊥ and show that W^⊥ is a subspace of V.
   
   
5. Let V be a vector space with inner product ⟨,⟩ and let W be a subspace of V. Define W^⊥ and show that W ∩W^⊥ = {0}.
   
   
6. Let V be a vector space with inner product ⟨,⟩ and let W be a subspace of V. Define W^⊥ and show that W+W^⊥= V.