I am at a very initial stage of commutative algebra.

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?

Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

Let $T:{\mathbb{R}}^3$ -> ${\mathbb{R}}^2$ be a linear transformation satisfying:
$T([1,0,1])=[2,3],T([2,1,3])=[-1,0],T([0,0,1])=[3,7]$
Find $T([x,y,z])$

Show that the system (G,⋅) is a group.

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle \lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}}$

Starting from the point ${\displaystyle (3,-2,-1)}$, reparametrize the curve ${\displaystyle x\left(t\right)=(3-3t,-2-t,-1-t)}$ in terms of arclength.

A simplified form of $(\sqrt{3^{2}x})+2+3^{-2x}$ that does not involve radicals or fractional exponents can be obtained after factoring the radicand. Find this simplification.