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assignment4

# assignment4 - → W be a linear transformation where V and...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences Linear Algebra II MATB24 Fall 2008 Assignment # 4 You are expected to work on this assignment prior to your tutorial in the week of October 6th, 2008. You may ask questions about this assignment in that tutorial. In your tutorial in the week of October 13th you will be asked to write a quiz based on this assignment and/or related material from the lectures and tutorials in week 4 and textbook readings. Textbook: Linear Algebra by Fraleigh & Beauregard, 3rd edition. Read: Chapter 3 Section 4 and Lecture 4 Notes Problems: 1. Fraleigh & Beauregard , Pages 227 #1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 34, 36, 37, 39, 42, 47, 48, 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
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Unformatted text preview: → W be a linear transformation where V and W are finite 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 ..., in a vector space V, define T: R n → V by v v , , 2 1 ( ) 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 ....
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