Determine whether the given set S is a subspace of the vector space V.A. V is the vector space of all real-valued functions defined on the interval [a, b], and S is the subset of V consisting of those functions satisfying f(a) = 4.OB. V is the space of three-times differentiable functions R→I and S is the subset of V consisting of those functions satisfying the differential equation y"" - y' = 1.OC. V = R"x", and S is the subset of all symmetric matricesR", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed m x n matrix.R², and S consists of all vectors (1, ₂) satisfying ² - x = 0.OF. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax³ + bx.OG. V is the space of five-times differentiable functions R → R, and S is the subset of V consisting of those functions satisfying the differential equation y(5)= 0.OD. V =OE. V

First noted that if be a vector space over a field be a subspace of ,then null vector of V must…

Answered: Determine whether the given set S is a…

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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Question

Determine whether the given set S is a subspace of the vector space V.
A. V is the vector space of all real-valued functions defined on the interval [a, b], and S is the subset of V consisting of those functions satisfying f(a) = 4.
OB. V is the space of three-times differentiable functions R→I and S is the subset of V consisting of those functions satisfying the differential equation y"" - y' = 1.
OC. V = R"x", and S is the subset of all symmetric matrices
R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed m x n matrix.
R², and S consists of all vectors (1, ₂) satisfying ² - x = 0.
OF. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax³ + bx.
OG. V is the space of five-times differentiable functions R → R, and S is the subset of V consisting of those functions satisfying the differential equation y(5)
= 0.
OD. V =
OE. V

Expert Solution

Step 1: vector spaces problem

First noted that if $V$ be a vector space over a field $F space a n d space S$ be a subspace of $V$ ,then null vector $theta$ of V must be in S.

A) Here $V equals open curly brackets f colon f colon open square brackets a comma b close square brackets rightwards arrow straight real numbers close curly brackets space$ be the set of all real valued functions,form a vector space .

Now given that $S equals open curly brackets f space element of V colon f left parenthesis a right parenthesis equals 4 close curly brackets$ .Noted that here zero function is the null vector of V,but $theta open parentheses a close parentheses equals 0 not equal to 4$ .

So $theta not an element of S$ .Hence S is not subspace of V.

B) Here ,V be the set of all three differentiable functions from $straight real numbers space t o straight real numbers$ .also $S equals left curly bracket y space element of V space colon y apostrophe apostrophe apostrophe minus y apostrophe equals 1 right curly bracket$

Noted that here zero function is the zero vector of V .but $theta apostrophe apostrophe apostrophe minus theta apostrophe equals 0 not equal to 1$ ,so we have $theta not an element of S$ .

Hence S is not subspace of V.

C) $V equals straight real numbers to the power of n cross times n end exponent comma$ is vector space of all real $n cross times n$ matrices and $S equals left curly bracket A element of V colon A to the power of T equals A right curly bracket$ .

noted that here zero matrix of order n is the zero vector of V.