6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
help with p.1.13 please. thank you! Group Name LAUSD Health N Vector Spaces P.1.9 Let V be an F-vector space, let wi, W2,...,W, EV, and suppose that at least one w; is nonzero. Explain why span{w1, W2,...,w,} = span{w; : i = 1,2,..., and W; 0). P.1.10 Review Example 1.4.8. Prove that U = {p EP3 : p(0) = 0) is a subspace of P3 and show that U = span{z.z.z). P.1.11 State the converse of Theorem 1.6.3. Is it...
just part c,d, and e please!! Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
Q5 (a) Provide the definition of the derivative of a map F: ViV2 where (V, l1) a are normed vector spaces (possibly infinite dimensional) (b) Let C((0, 1) be the space of continuous real valued functions on [0, 1] endowed with the supremum norm. Define F:C ((0, 1]) C([0, 1]) by F(() Jo f()dt, e for all f E C(0, 1). Show directly from the definition that the derivative of F is differentiable on the entire domain. (c) For the...
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
VECTOR SPACES LINEAR ALGEBRA Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2): u + v = (u1 + v1 + 1, u2 + v2 + 1), ku = (ku1, ku2) a) Show that (0,0) does not = 0 b) Show that (-1, -1) = 0 c) Show that axiom 5 holds by producing an ordered pair -u...
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector space" and "subspace" as precisely as you can. (b) Consider a line through the origin in R2, for example, the r-axis. Explain why this line is, or is not, a subspace of R2 in terms of your definitions in (a). (c) Consider the union of two lines through the origin in R2, for example, the z- and y-axes. Explain why this union of lines...
Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let V be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v EV : there is w E Wo such that T(v) = w} If T is injective then dim(VO) = dim(W). dim(ker(T) n Vo) = dim(VO) - dim(Wo).
Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т € R?. (1x2)f(x)|d for f(x) e C[0, 1] (i) _ (iii) pl dо + 2la| + 3/az| for p(z) — аz2? + ајя + ao є Pз. Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т €...