Which of the following sets are linear spaces?

a.) {X = (x1, x2, x3) in R3 with the property x1-2x3=0}

b.) The set of solutions of x= of Ax= =0, where A is an m*n matrix.

c.) The set of 2x2 matrices A with det(A) =0.

d.) The set of polynomials p(x) with the definite integral from (-1) to (1) of P(x)dx=0.

e.) The set of solutions y=-y(t) of y'' +4y'+y=0.

Please explain each option as best as possible.

Question

Which of the following sets are linear spaces?

a.)    {X = (x1, x2, x3) in R3 with the property x1-2x3=0}

b.)    The set of solutions of x=> of Ax=> =0, where A is an m*n matrix.

c.)     The set of 2x2 matrices A with det(A) =0.

d.)     The set of polynomials p(x) with the definite integral from (-1) to (1) of P(x)dx=0.

e.)     The set of solutions y=-y(t) of y'' +4y'+y=0.


Please explain each option as best as possible.