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My answers aren't all correct and I am not sure why..

Problem: 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)=f(b).

B. 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.

C. V=C^1(R), and S is the subset of V consisting of those functions satisfying f′(0)≥0.

D. V=P_4, and S is the subset of P_4 consisting of all polynomials of the form p(x)=a(x^3)+(bx).

E. V=R^n×n, and S is the subset of all symmetric matrices.

F. V=R^4, and S is the set of vectors of the form (0,x2,3,x4).

G. V=R^n×n, and S is the subset of all matrices with det(A)=0.

I chose A,B,C, E, G.

A: sine an cosine are real valued functions. in the case of sine,on the interval [0,2pi] f(0)=f(2pi).

B:Just looking at the dimensions A has n columns which means( i think) n unknowns therefore falls in R^n

C: if f(t) = t then f'(t)=1 which is greater than 0 at t=0.

E:a symmetric matrix is a square matrix

G:A must be square in order to take a determinate.

I did not choose D and F:

D: because P(x) is of degree 3 and V=P_4

F: does not contain the 0 vector.

Any insight would be greatly appreciated.

Problem: 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)=f(b).

B. 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.

C. V=C^1(R), and S is the subset of V consisting of those functions satisfying f′(0)≥0.

D. V=P_4, and S is the subset of P_4 consisting of all polynomials of the form p(x)=a(x^3)+(bx).

E. V=R^n×n, and S is the subset of all symmetric matrices.

F. V=R^4, and S is the set of vectors of the form (0,x2,3,x4).

G. V=R^n×n, and S is the subset of all matrices with det(A)=0.

I chose A,B,C, E, G.

A: sine an cosine are real valued functions. in the case of sine,on the interval [0,2pi] f(0)=f(2pi).

B:Just looking at the dimensions A has n columns which means( i think) n unknowns therefore falls in R^n

C: if f(t) = t then f'(t)=1 which is greater than 0 at t=0.

E:a symmetric matrix is a square matrix

G:A must be square in order to take a determinate.

I did not choose D and F:

D: because P(x) is of degree 3 and V=P_4

F: does not contain the 0 vector.

Any insight would be greatly appreciated.

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