Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, x, y

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

How would one go about analytically solving a system of non-linear equations of the form:
$a+b+c=4$
$a^2+b^2+c^2=6$
$a^3+b^3+c^3=10$
Thank you so much!

Physical meaning of the null space of a matrix
What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank r of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an r-dimensional space, where r is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?

Consider the system (*) whose coefficient matrix A is the matrix D listed in Exercise 46 and whose fundamental matrix was computed just before the preceding exercise.

$x+y+2z=3$
$3x+2y+2z=4$
$x+y+3z=5$

Reduce the system of linear equations to upper triangular form and solve.
5x+2y=8
-x+3y=9

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or $x_1,x_2,x_3,x_{4}x$ ​ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{cccccc}1& 0& 0& 0& |& 1\\ 0& 1& 0& 0& |& 2\\ 0& 0& 1& 2& |& 3\end{array}\right]$