The reduced row echelon form of the augmented matrix of a system of linear eqs. is given. Determine whether this system of linear eqs. is consistent

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

Write the vector form of the general solution of the given system of linear equations. ${\displaystyle x_1-x_2-2x_4-x_5+4x_6=0}$

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. $\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 0& 3\\ 0& 0& 1& 4\\ 0& 0& 0& 0\end{array}\right]$

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 $x_1,x_2\dots$ from left to right. $[1,2,0,2,-1,3]$

Write the system of linear equations in the form ${\displaystyle Ax=b}$ and solve this matrix equation for x. ${\displaystyle -x_1+x_2=4}$
${\displaystyle -2x_1+x_2=0}$

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