University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 14: Due December 12, 2007

1. Review the page on How to do proofs: and download, print out and stare at
   Fields and Vector spaces: pdf file
   Fields and Vector spaces, the proofs: pdf file
   .
   
   
2. Define the following terms:
• vector space
• subspace
• span(S)
• linear combination
• linearly independent
• basis
• linear transformation
• ker L
• im L
• eigenvector, eigenvalue and eigenspace
• inner product
• orthogonal
• orthonormal basis
• W^⊥
  
  
3. Let A be an n × n matrix. Prove that A is invertible if and only if det(A) is nonzero.
   
   
4. Prove that det(AB)=det(A)det(B).
   
   
5. Let V be a vector space and let S = {s₁,... s_n} be a subset of V. Show that span(S) is the set of linear combinations of s₁, ..., s_n.
   
   
6. Let V be a vector space and let L:V → V be a linear transformation. Show that the λ eigenspace of V is the kernel of the linear transformation L-λ.
   
7. 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.