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tut4 - MATH 1104F Solutions to Tutorial Assignment 4 Total...

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MATH 1104F - Solutions to Tutorial Assignment 4 February 23, 2010 Total: 10 points 1. Let S = { ( x, y ) R 2 : x 0 and y 0 } be a subset of R 2 . Is S a subspace of R 2 ? Why or why not? Solution: S is not a subspace of R 2 because it is not closed under scalar multiplication. For example, if ( x, y ) S with x > 0 and y < 0 and if r < 0 then r ( x, y ) = ( rx, ry ) is not in S as rx < 0 (or ry > 0). 2. Let T : R 2 R 2 be the linear transformation defined by T ( x, y ) = (3 x + y, 2 x + y ) . Find T - 1 ( x, y ). Solution: The standard matrix for T is A = 3 1 2 1 . The inverse of A is the standard matrix for T - 1 . A - 1 = 1 det A 1 - 1 - 2 3 = 1 3 · 1 - 2 · 1 1 - 1 - 2 3 = 1 - 1 - 2 3 Hence A - 1 · x y = 1 - 1 - 2 3 · x y = x - y - 2 x + 3 y and thus T - 1 ( x ) = ( x - y, - 2 x + 3 y ) . 3. Let A = 1 2 5 0 - 1 0 1 4 0 1 1 2 5 1 - 4 1 3 9 2 - 6 . (a) Find p and q for which Nul A is a subspace of R p and Col A is a subspace of R q . (b) Find a basis for Nul A and a basis for Col A . (c) Find a nonzero vector in Nul A and a nonzero vector in Col A .

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tut4 - MATH 1104F Solutions to Tutorial Assignment 4 Total...

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