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HW 11 8 – 10 Let CR denote the set of continuous functions on R (this is a vector space using function addition and scalar multiplication). Define L : CR → FR by Z x L(f )(x) = f (t) dt. 0 Show that L is linear. 8 – 11 Let S be the subset of FR consisting of all functions f for which f (−x) = f (x) for all x ∈ R. Show that S is a subspace of FR . Homework List 8 – 12 Let L : M2×2 → P2 be the linear function given by a b L = (a − d)x + (b + c). c d ◭◭ ◮◮ ◭ ◮ (a) Find ker L. Page 1 of 3 (b) Find im L. Back Home Page 8 – 13 Let L : V → V ′ be a linear function, let a ∈ V ′ and let vp be a particular vector in V such that L(vp ) = a. (a) Show that if v0 is in ker L, then L(vp + v0 ) = a. (b) Show that if v is a vector in V such that L(v) = a, then v = vp + v0 for some v0 ∈ ker L. (Hint: If you show that v − vp ∈ ker L, then you can conclude that v − vp = v0 for some v0 ∈ ker L.) (c) Recall that L−1 (a) is the set of all v in V such that L(v) = a. Show that L−1 (a) = {vp + v0 | v0 ∈ ker L}. (Hint: Use parts (a) and (b) to show that each set is contained in the other.) 8 – 14 Let L : DR → FR be the function given by L(y) = y ′ + 3y Homework List ◭◭ ◮◮ ◭ ◮ (DR is the vector space of differentiable functions). (a) Show that L is linear. (b) Let yp be a particular solution to the linear differential equation y ′ + 3y = 1. Show that every solution of y ′ + 3y = 1 is of the form y = yp + y0 where y0 is a solution to the corresponding homogeneous equation y ′ + 3y = 0. (Hint: Use part (a) and Exercise 8 – 13(b).) Page 2 of 3 Back Home Page 8 – 15 8 – 16 5 6 In the vector space M2×2 , determine whether is in 1 −5 1 −3 −2 6 2 1 Span{ , , }. −4 −1 8 2 −1 −2 Let 4 8 3 , g(x) = 2 − , h(x) = 1 + . x x x Determine whether the functions f, g, and h are linearly independent in the vector space FR+ , where R+ is the set of positive real numbers. f (x) = 1 + Hint: At some point, make three choices for x to get a system of three equations. Homework List ◭◭ ◮◮ ◭ ◮ Page 3 of 3 Back Home Page