2. Let F = {f : R → R} be the vector space of functions from R to R. a. Decide which of the following subsets of F is a vector subspace of F. In each case, either give a proof that it is a subspace or demonstrate an axiom that is not satisfied. Gi = {g : R → R : g(x) = –9(-x) for all x E R}, G2 = {g : R → R: g(n+1) = g(n) +1 for all n E Z}. %3D b. For each n > 1, define a function f, E F such that for all k > 1 the functions f1, f2, f3, ... , fk-1, fk are linearly independent.

2. Let F = {f : R → R} be the vector space of functions from R to R. a. Decide which of the following subsets of F is a vector subspace of F. In each case, either give a proof that it is a subspace or demonstrate an axiom that is not satisfied. Gi = {g : R → R : g(x) = –9(-x) for all x E R}, G2 = {g : R → R: g(n+1) = g(n) +1 for all n E Z}. %3D b. For each n > 1, define a function f, E F such that for all k > 1 the functions f1, f2, f3, ... , fk-1, fk are linearly independent.

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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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

2. Let F = {f : R → R} be the vector space of functions from R to R.
a. Decide which of the following subsets of F is a vector subspace of F. In each case, either give a
proof that it is a subspace or demonstrate an axiom that is not satisfied.
Gi = {g : R → R : g(x) = –9(-x) for all x E R},
G2 = {g : R → R: g(n+1) = g(n) +1 for all n E Z}.
%3D
b. For each n > 1, define a function f, E F such that for all k > 1 the functions
f1, f2, f3, ... , fk-1, fk
are linearly independent.

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