Answered: Problem 43. Let L: V → W be a linear…

Problem 43. Let L: V → W be a linear function and define def Graph(L) {(v, L(v)) E V x W : vEV}. (a) Prove that Graph(L) is a vector subspace of V × W.

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Problem 43. Let L: V → W be a linear function and define
def
Graph(L) {(v,L(v)) E V x W : veV}.
(a) Prove that Graph(L) is a vector subspace of V x W.
(b) Prove that V =Graph(L).
(c) Prove that V x W/Graph(L) =w.

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