Question

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

Forms of linear equations
ANSWERED
asked 2021-06-10
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 p(1)>p(0).
B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).
C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=\(C^2(I)\), 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=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.
G. \(V=M_n(R)\), and S is the subset of all symmetric matrices

Answers (2)

2021-06-11
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Best answer
2021-09-09

Solution.

A subset S is a subspace of V if \(\alpha x+\beta y\in S\) for all x, \(y\in S\) and \(\alpha,\beta \in R\)

A) Not a subspace

\(p(x)=x^2+5,\ g(x)=x^2-5\)

\(p(1)=6,\ p(0)=5\) and \(p(1)>p(0)\)

\(q(1)=-4,\ q(0)=-5\) and \(q(1)>q(0)\)

\(\Rightarrow p(x),q(x)\in S\)

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 \(R(1)<R(0)\)

\(\Rightarrow R(x)\ne\in S\)

\(\Rightarrow S\) is not a subspace

B) Not a subspace

\((-1,-1,0)\in S\) as \((-1)-(-1)6+0=-1+6=5\)

Consider \(-1(-1,-1,0)=(1,1,0)\)

\(1-6+0=-5\ne 5\)

\(\Rightarrow(1,1,0)\ne \in S\)

\(\Rightarrow S\) is not subspace

C) Subspace

Let \(x,y\in S\) then \(Ax=0\) and \(Ay=0\)

\(A(\alpha x+\beta y)=\alpha Ax+\beta Ay\)

\(=\alpha(0)+\beta(0)\)

\(=0\)

\(\Rightarrow\alpha x+\beta y\in S\)

\(\Rightarrow S\) is a subspace

D) Subspace

Let \(x,y\in S\) then \(x''-4x'+3x=0\)

and \(y''-4y'+3y=0\)

Consider \(\alpha x+\beta y\)

\((\alpha x+\beta y)''-4(\alpha x+\beta y)'+3(\alpha x+\beta y)\)

\(=\alpha x''+\beta y''-4\alpha x'-4\beta y'+3\alpha x+3\beta y\)

\(=\alpha(x''-4x'+3x)+\beta(y''-4y'+3y)\)

\(=\alpha(0)+\beta(0)\)

\(=0\)

\(\Rightarrow\alpha x+\beta y\in S\)

\(\Rightarrow S\) is a subspace

E) Not a subspace

Let \(f,g\in S\) then \(f(a)=5\) and \(g(a)=5\)

\(\alpha f(a)+\beta g(a)=5\alpha+5\beta\)

Let \(\alpha=1\) and \(\beta=-1\)

\(\Rightarrow\alpha f(a)+\beta g(a)=5-5=0\)

\(\Rightarrow\alpha f(a)+Bg(a)\ne\in S\)

\(\Rightarrow S\) is not a subspace

F) Subspace

Let \(p,g\ne\in S\) then \(p(0)=0\) and \(g(0)=0\)

Consider \(\alpha p+\beta g\)

\(\alpha p(0)+\beta q(0)=\alpha(0)+\beta(0)=0\)

\(\Rightarrow\alpha x+\beta y\in S\)

\(\Rightarrow S\) is a subspace

G) Subspace

Let \(A,B\in S\) then \(A^T=A\) and \(B^T=B\)

Consider \(\alpha A+\beta B\)

\((\alpha A+\beta B)^T=(\alpha A)^T+(\beta B)^T\)

\(=\alpha A^T+\beta B^T\)

\(=\alpha A+\beta B\)

\(\Rightarrow\alpha x+\beta y\in S\)

\(\Rightarrow S\) is a subspace

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