Determine whether the given set S is a subspace of the vector space V. A. V=P_5, and S is the subset of P_5 consisting of those polynomials satisfying

Efan Halliday 2021-06-10 Answered
Determine whether the given set S is a subspace of the vector space V.
A. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
B. V=R3, and S is the set of vectors (x1,x2,x3) in V satisfying x16x2+x3=5.
C. V=Rn, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=C2(I), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. 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)=5.
F. V=Pn, and S is the subset of Pn consisting of those polynomials satisfying p(0)=0.
G. V=Mn(R), and S is the subset of all symmetric matrices
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diskusje5
Answered 2021-06-11 Author has 82 answers

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Jeffrey Jordon
Answered 2021-09-09 Author has 2495 answers

Solution.

A subset S is a subspace of V if αx+βyS for all x, yS and α,βR

A) Not a subspace

p(x)=x2+5, g(x)=x25

p(1)=6, p(0)=5 and p(1)>p(0)

q(1)=4, q(0)=5 and q(1)>q(0)

p(x),q(x)S

Consider R(x)=p(x)2q(x)

R(1)=p(1)2q(1)=6+8=24

R(0)=p(0)2q(0)=5+10=25

Here R(1)<R(0)

R(x)≠∈S

S is not a subspace

B) Not a subspace

(1,1,0)S as (1)(1)6+0=1+6=5

Consider 1(1,1,0)=(1,1,0)

16+0=55

(1,1,0)≠∈S

S is not subspace

C) Subspace

Let x,yS then Ax=0 and Ay=0

A(αx+βy)=αAx+βAy

=α(0)+β(0)

=0

αx+βyS

S is a subspace

D) Subspace

Let x,yS then x4x+3x=0

and y4y+3y=0

Consider αx+βy

(αx+βy)4(αx+βy)+3(αx+βy)

=αx+βy4αx4βy+3αx+3βy

=α(x4x+3x)+β(y4y+3y)

=α(0)+β(0)

=0

αx+βyS

S is a subspace

E) Not a subspace

Let f,gS then f(a)=5 and g(a)=5

αf(a)+βg(a)=5α+5β

Let α=1 and β=1

αf(a)+βg(a)=55=0

αf(a)+Bg(a)≠∈S

S is not a subspace

F) Subspace

Let p,g≠∈S then p(0)=0 and g(0)=0

Consider αp+βg

αp(0)+βq

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