Fix a field F. The direct sum of two F-vector spaces V and W is the F-vector spaceVW defined as follows. As a set, V W is equal to V x W, and addition and scalarmultiplication on VW are defined component-wise: (v₁, W₁) +(v2, W₂) = (v₁+V2₂, W₁+W₂)and c(v, w) = (cv, cw).Problem 14. Prove that a function f: V → W is linear if and only if its graph¹ is a subspaceof VW.

Assume f:V→W is linear. We need to prove that the graph G of f, G={ (v,f(v)) ; v∈V} is a subspace of…

Answered: Problem 14. Prove that a function f: V→…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1 10. Find the dimension of the subspace V = span{v1, v2, V3, v4} in R³ where vi = 1 2 1 V2 = 1 V3 = and v4 = 1 2 1 1
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Problem 4. Explain whether or not the following sets are subspaces of the given vector space (a) : a, bɛR of R³? (b) :a € of R? (c) {p(t) : p(1) = 1} of P2?
4. Let VCR" be a subspace. Let F: V→V and G: V → V be invertible linear transformations. Denote by F¹: V→V and G¹: V → V the inverses of F and G respectively. Is the map H: V → V defined by H(v):= F(G(F¹(G¹(v)))), for v € V, a linear transformation? Justify your answer.
1 4. Consider the vectors v1 , Ú2 , U3 1 -2 1 Are they linearly independent? If not either: (a) Find a linear dependence relation among them. or: (b) Express one of the vectors as a linear combination of the others.
(c) Let C be the vector space of continuous real-valued functions on R. If T: C → C is defined so T(f(x)) = |ƒ (x)\, is T a linear transformation? Explain why or why not.
-(.. Suppose that T: V V is a linear transformation and V is a finite dimensional vector space. Suppose that U is a subspace of V such that T(U) = T(V). Prove that U + ker(T) = V. %3D
3 1. Consider the subspaces of V = R³ below, Find U + W, U + Z, and W + Z. U = W = Z = {(x, y, x − y) : x, y ≤ R} {(x, 0, x) : x ≤ R} {(x, x, x) : x ≤ R}
(b) If f is a bounded linear functional on a Hilbert space H, show that there is a unique y e H such that f(x) = for all ХE Н.
Let V and W be vector spaces and let T : V → W be a linear transforma- tion. Suppose that {T(v₁), T(v₂),..., T(vn)} is linearly independent. Prove that {V₁, V2,..., Vn} is linearly independent.

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Problem 14. Prove that a function f: V→ W is linear if and only if its graph¹ is a subspace of VW.