# sol-a4 - University of Toronto at Scarborough Department of...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer and Mathematical Sciences Linear Programming and Optimazation MATB61 Winter 2008 Solution set to Assignment 4 Addition: 1) a)Yes. dim(range(T)) = dim(V) – dim(ker(T)) = 5 – 2 = 3 = dim(W). since range(T) W, range(T) = W by section 3.2 #36. ⊂ Therefore, T is onto. b) No. For example: T: R 2 → R 2 by T(x, y) = (y, 0). ker(T) = range(T). 2) a) ker(T) = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∈ = + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ R d c b a d a d c b a , , , , = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∈ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − R c b a a c b a , , = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∈ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − R c b a c b a , , 1 1 1 1 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 1 , 1 , 1 1 sp A basis of ker(T) is ....
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## This note was uploaded on 12/20/2010 for the course MAT MATB24 taught by Professor X.jiang during the Spring '10 term at University of Toronto- Toronto.

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sol-a4 - University of Toronto at Scarborough Department of...

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