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Mathematics 369 Homework (due Feb 1) A. Hulpke 7) Let V be a vector space over F and 0 ∈ V the zero vector. Show that for every c ∈ F we have that c · 0 = 0. 8) Let x, y, z be elements of a vector space V . Show that if x + y = x + z then y = z. 9∗ ) Let V = {(a : b) | a, b ∈ R} the set of all colon-separated pairs of real numbers. We define an “addition” (this is not the usual componentwise addition, we just call it “addition” for this problem) on V by setting (a : b) + (c : d) = (a + c) : (b − d) and a scalar “multiplication” by R on V by α · (a : b) = (αa : αb). Is V with these operations a vector space? Justify your answer! 10) Which of the following sets are vector spaces over R? Give a proof or a counterexample (Avoid unnecessary work. Be careful which axioms you actually need to check!)? a) The set of even functions: { f : R → R | f (−x) = f (x)} (with the usual scalar multiplication and addition of functions). b) The set C of complex numbers with the usual addition and the usual scalar multiplication by R. c) The set of ordered 2-tuples {(a, b)T ∈ R2 | a ≥ b} with componenwise addition and scalar multiplication by R. d) The set of solutions to the differential equation f 00 (x) + f 0 (x) − 3 f (x) = 0. e) The set {(0, 0)T } with componentwise addition and scalar multiplication. f) The set of nonsingular matrices {A ∈ R2×2 | det(A) 6= 0}. 11) In each of the following cases you are given a vector space V (all of them have been proven to be a vector space in class) and a subset S ⊂ V . In each case, check whether S is a subspace of V (in which case we can write S ≤ V ). (Be aware that you only need to check the closure axioms!) Justify your answer. a) V = R3 , S = {(1, a, b)T ∈ V | a, b ∈ R}. b) V = R3 , S = {(a, b, c)T ∈ V | 3a + 4b = 2c}. c) V = R3 , S = {(a, b, c)T ∈ V | a2 = 1}. d) V = C[0, 10], S = { f ∈ V | f (5) = 0}. e) V = C[0, 10], S = { f ∈ V | f (5) = 1}. f) V = C2 [−∞, ∞], S = { f ∈ V | f 00 + 3 f 0 = 0}. g) V = P3 , S = { f ∈ V | f (x) = a + bx2 } (i.e. f is a polynomial without an x-term). Problems marked with a ∗ are bonus problems for extra credit. Set Notation In mathematics one often wants to define sets of elements, by selecting some elements from a larger set A, that fulfill a certain property. (For example: All positive real numbers that are smaller than 5.) This is done easiest by the following notation: {a ∈ A | property(a)} The left part (before the |) indicates that we take elements from a set A. The name a is only used, as we want to refer to these elements in the right part. The right part indicates a property, that these elements will have to fulfil, i.e. the new set only consists of those elements, for which the property is fulfilled. For example, I could describe the set of students with a blue sweater as: S1 = {s ∈ Students | s wears a blue sweater} If R+ is the set of positive real numbers we can denote the positive real numbers smaller than 5 by S2 = {a ∈ R+ | a < 5} or I could just refer to the real numbers R and impose two conditions: S2 = {a ∈ R | a > 0, a < 5} Similarly, the set S3 = { f ∈ C[0, 2] | f (1) = 0} consists of all functions continuous on the interval [0, 2] that have a zero at 1. This notation is not unique. We could describe S2 as: {a ∈ R | a > 0, a < 5} or {a ∈ R | 0 < a < 5}. When working with such sets, one typically will use the specified property. For example if we use again S3 = { f ∈ C[0, 2] | f (1) = 0}, we can deduce that f (1) = 0 (which definitevely does not hold for any arbitrary function). On the other hand, if g is an arbitrary function, we can check that g ∈ S3 by verifying the property, i.e. that g(1) = 0. Thus for example x − x2 ∈ S3 and ex 6∈ S3 . (Some people prefer to use a semicolon (;) instead of the line |.)