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Math 110: Tutorial Problems — Week 13 1. Which of the following transformations are linear? Justify your claim. x x+y x xy x 2x − 3y T1 = , T2 = , T3 = , y y+1 y y y 2x − 3y T4 x y = x3 x2 . 2. Let H be the plane in R3 defined by x + y + z = 0. Find the standard matrix of the projection map PH : R3 → R3 onto H. What is the matrix of the map SH : R3 → R3 which reflects each vector in H? x x + 2y x x+y 3. Let T = and S = . y x+y y x+y (a) Find T ◦ S and S ◦ T . (b) Is T invertible? Is S invertible? For each map which is invertible, find its inverse. Math 110: Tutorial Problems — Week 14 1. In the following determine whether the set W is a subspace of the vector space V . a 3 −a 1. V = R , W = |a∈R 2a a b 2. V = M2,2 the vector space of 2×2 matrices with entries in R, W = | ad ≥ bc . c d 2. Is M2,2 spanned by 1 1 0 1 0 1 1 0 0 −1 , , and ? 1 0 1 1 1 0 3. Prove that every vector space has a unique zero vector. Math 110: Tutorial Problems — Week 15 1 0 1 2 1 2 1 −2 1. Let u1 = , u2 = , u3 = , u4 = ∈ M2 (R). Find 1 0 1 0 4 2 1 0 the E-coordinates of u1 , . . . , u4 and use these to determine whether S1 := {u1 , u2 , u3 } and S2 = {u1 , u2 , u4 } are linearly independent. Also, find dim(span(Si )), for i = 1, 2. 2. Let ~v1 , . . . , ~vn , ~u ∈ V be vectors in a vector space V . Prove: (a) span(~v1 , . . . , ~vn , ~u) = span(~v1 , . . . , ~vn ) if and only if ~u ∈ span(~v1 , . . . , ~vn ). (b) If ~v1 , . . . , ~vn are linearly independent, then ~v1 , . . . , ~vn , ~u are linearly independent if and only if ~u ∈ / span(~v1 , . . . , ~vn ). 3. Let {~u, ~v , w} ~ be a linearly independent set of vectors in a vector space V , and let S1 = {~u + ~v , ~v + w, ~ ~u + w} ~ and S2 = {~u − ~v , ~v − w, ~ ~u − w}. ~ (a) Is S1 linearly independent? (Justify!) Find dim(W1 ), where W1 = span(S1 ). (b) Is S2 linearly independent? (Justify!) Find dim(W2 ), where W2 = span(S2 ).