MAT444test2-2009-soln

# MAT444test2-2009-soln - MAT 442: Test 2 Solutions...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT 442: Test 2 Solutions Throughout, F is a field. 1. Read these instructions carefully. For each type of row operation give the following: (a) A description of what it does (so we know which row operation you are talking about) (b) The determinant of a corresponding elementary matrix E (c) State whether or not E t is an elementary matrix (d) State whether or not E is invertible, and how we know that. Type I operations: a) Swap row i with row j with i 6 = j . b) det( E ) =- 1 c) Yes it is an elementary matrix, in fact, E t = E d) Yes it is invertible because we can undo the operation by repeating it: E 2 = I Type II operations: a) Multiply row i by k ∈ F where k 6 = 0 b) det( E ) = k c) Yes it is an elementary matrix, in fact, E t = E d) Yes, because we can undo the operation by multiplying the same rwo by 1 /k , which exists since k 6 = 0 Type III operations: a) Add k times row i into row j where i 6 = j (i.e., R j ← R j + kR i ) b) det( E ) = 1 c) Yes it is an elementary matrix – the transpose corresponds to adding k times row j into row i d) Yes it is invertible because we can undo the operation by adding- k times row i into row j , which corresponds to multiplying by another elementary matrix 2. (a) Define eigenvalue and eigenvector If T is an operator on a vector space V over F , an eigenvector x ∈ V is a non-zero vector such that T ( x ) = λx for some λ ∈ F . A scalar λ is an eigenvalue if there is a corresponding eigenvector x as above. (b) Define eigenspace If λ ∈ F is an eigenvalue of a linear operator T on a vector space V , the λ- eigenspace is the set { v ∈ V | T ( v ) = λv } . (c) Define characteristic polynomial, including an explanation as to why the definition...
View Full Document

## This document was uploaded on 09/25/2011.

### Page1 / 4

MAT444test2-2009-soln - MAT 442: Test 2 Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online