Problem 1 10 pts Let T : V ! V and S : V ! V be two linear...

• Test Prep
• 14

This preview shows page 1 - 6 out of 14 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 7 / Exercise 64
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
Tan
Expert Verified
Problem 1 (10 pts) Let T : V ! V and S : V ! V be two linear transformations. Show that ST is invertible if and only if S and T are invertible. 2
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 7 / Exercise 64
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
Tan
Expert Verified
Problem 2 (10 pts) Let V, h , i be a real inner product space. Let W be a subspace of V . Define W ? to be the space of all x 2 V such that h x, y i = 0 for all y 2 W i.e: W ? = { x 2 V : h x, y i = 0 for all y 2 W } 1. Prove that W ? is again a subspace of V . 2. Prove that W ? \ W = { 0 } . 3
3. Let x 1 , ..., x k be an orthonormal basis of W . Define T : V ! V by T ( x ) = k i =1 h x, x i i x i . Then T is a linear operator (you may use this without proving it). Prove that the range of T is W and the kernel of T is W ? . 4
4. Prove that dim ( W ) + dim ( W ? ) = dim ( V ) . 5
Problem 3 (10 pts) A matrix U in M n n ( R ) is said to be unitary if U t U = I n 1. Assume that U is unitary and λ is an eigenvalue of U . Prove that λ is 1 or - 1. 2. Give an example of a unitary matrix in M 2 2 ( R ) which does not have a real eigenvalue. 6