The coefficient matrix for a system of linear differential equations of the form y^1=Ay

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

In the matrix of a system of linear equations, suppose that two of the rows are equal. What can you say about the rowreduced form of the matrix?

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $\left[\begin{array}{ccccc}1& 0& -1& |& 1\\ 0& 1& 2& |& 1\end{array}\right]$

Determine whether the following statement is true or false. If the reduced row echelon form of the augmented matrix of a consistent system of mm linear equations in nn variables contains kk nonzero rows, then its general solution contains kk basic variables.

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $\left[\begin{array}{cccccc}1& 0& -1& -2& -3& 0\\ 0& 1& 2& 3& 4& 0\end{array}\right]$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2,...$ form left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

Determine whether the given $(2\times 3)(2\times 3)$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$x=at+b,y=ct+d,z=et+f$
$2x_1+x_2+x_3=3$
$-2x_1+x_2-x_3=1$