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Mathematics 369 Homework (due Mar 1) A. Hulpke 25) Which of the following functions R2 → R3 are linear? Justify your answer. a) (x, y)T 7→ (0, x − y, x + y)T b) (x, y)T 7→ (xy, 0, (x + 1)(y + 1))T c) (x, y)T 7→ (x + 1, y + 1, 3)T d) (x, y)T 7→ (−y, −x, 3x − 7y)T (This problem was already on sheet 5. It has been bumped one week back.) 29) Let V = C1 [0, 1] be the vector space of functions on [0, 1] that are differentiable. Which of the following mappings are linear? Justify your answer: a) The differentiation mapping D:V → V , f 7→ f 0 that assigns to each function its derivative. b) The integration mapping I:V → V that assigns to each f an antiderivative F such that F(0) = 0 and F 0 = f . c) The evaluation map E:V → R, f 7→ f (5). d) The (argument) shift map S:V → V , f (x) 7→ f (x + 1). e) The (value) shift map T :V → V , f (x) 7→ f (x) + 1. 30) Is there a linear map from R2 → R2 that maps the vectors x = (1, 0)T , y = (1, 2)T , z = (3, 4)T as prescribed? Explain. (Hint: The set {x, y} is a basis. We can always prescribe a linear map on a basis.) a) x 7→ (2, 3)T , y 7→ (0, 1)T , z 7→ (1, 5)T . b) x 7→ (2, 3)T , y 7→ (0, 1)T , z 7→ (2, 5)T . c) x 7→ (1, 1)T , y 7→ (1, 1)T , z 7→ (3, 3)T . 31∗ ) Find an example of two linear maps L, M: R2 → R2 such that L(M(x)) = 0 for all x ∈ R2 , but that there exists an element y ∈ R2 such that M(L(y)) 6= 0. 32) Let b1 = 1 + x, b2 = 1 − x and V = Span(b1 , b2 ) ≤ P2 . Then B = (b1 , b2 ) is a basis of V (you do not need to show this). Consider the transformation L:V → V , defined by L: f (x) 7→ x · f 0 (x). Then L is linear (you do not need to show this either). a) Determine [L(b1 )]B and [L(b2 )]B . b) Determine B [L]B . Problems marked with a ∗ are bonus problems for extra credit.