If A is an n x n matrix , where are the entries on the main diagonal of A-A^T? Justify yoyr answer.

Find an explicit description of Nul A by listing vectors that span the null space.
$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

Assume that A is row equivalent to B. Find bases for Nul A and Col A.
$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$
$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$
Find a nonzero vector in Nul A.
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Find the dimension of each of the following vector spaces. (a) The vector space of all diagonal ${\displaystyle n\times n}$ matrices. (b) The vector space of all symmetric ${\displaystyle n\times n}$ matrices. (c) The vector space of all upper triangular ${\displaystyle n\times n}$ matrices.

I'd like to be able to enter a vector or matrix, see it in 2-space or 3-space, enter a transformation vector or matrix, and see the result. For example, enter a 3x3 matrix, see the parallelepiped it represents, enter a rotation matrix, see the rotated parallelepiped.

Let $K=\mathbb{R}$ and $V:=\mathbb{R}[t{]}_2$ and let $F:\mathbb{R}[t{]}_2\to \mathbb{R}[t{]}_2,f\mapsto f(0)+f(1)\cdot (t+t^2)$
Want to calculate $M_S^S(F)$, the transformation Matrix of F with Basis S, being the standardbasis for the polynomialring $S=(1,t,t^2)$
So we have to use the coordinate isomorphism:
$M_S^S(F)=\left(\begin{array}{ccc}{I}_S(F(1))& I_S(F(t))& I_S(F(t^2))\end{array}\right)$
Already know the solution, however I have no idea why the following matrix is the transformation Matrix of F.
$M_S^S(F)=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$
Looking at the transformation matrix we know that $F(1)=1+t+t^2$?

Suppose that $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation. Prove that there exists an $m\times n$ matrix $A$ such that for all $x\in {\mathbb{R}}^n$, $T(x)=Ax$. In other words, $T$ is the "matrix transformation" associated with $A$.