assignment5 - dimensional a If dim(V = 5 dim(W = 3 and...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences Linear Algebra II MATB24 Fall 2010 Assignment # 5 You are expected to work on this assignment prior to your tutorial in the week of October 19th, 2010. You may ask questions about this assignment in that tutorial. In your tutorial in the week of October 26th you will be asked to write a quiz based on this assignment and/or related material from the lectures and tutorials in week 5 and textbook readings. Textbook: Linear Algebra by Fraleigh & Beauregard, 3rd edition. Read: Chapter 3 Section 4 and week 5 Lecture Notes Problems: 1. , Pages 227 # 11, 12, 13, 34, 36, 37, 39, 42, 47, 49, 50 2. Addition: 1) In each case either prove the statement or give an example in which it is false. Throughout, let T: V → W be a linear transformation where V and W are finite
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Unformatted text preview: dimensional. a. If dim(V) = 5, dim(W) = 3, and dim(ker(T))= 2, then T is onto. b. If ker(T) = V, then W = { 0 }. 2) In each case, (i) find a basis of ker(T), and (ii) find a basis of range(T) a. T: M 22 → R; d a d c b a T ; b. T: P 2 → R 2 ; T(a + bx + cx 2 ) = (a, b). 3) Let T: M nn → R denote the trace map: T(A) = tr(A) for all A in M nn . Show that dim(ker(T)) = n 2 – 1. (hint: dimension theorem) 4) Given n v v v ..., , , 2 1 in a vector space V, define T: R n → V by n n n v r v r v r r r r T ... ..., , , 2 2 1 1 2 1 . Show that T is linear, and that: a. T is one-to-one if and only if n v v v ..., , , 2 1 is independent. b. T is onto if and only if V = sp n v v v ..., , , 2 1 . Note: There are answers at the back of the textbook for the odd number questions....
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This note was uploaded on 12/25/2010 for the course MATHEMATIC MATB24 taught by Professor Yang during the Fall '09 term at University of Toronto.

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