Find the matrix of the linear mapping. L(x, y, z)=(x+y+z)

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

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.
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

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

Why is the Jacobian matrix equal to the matrix associated to a linear transformation?
Given the linear transformation f, we can construct the matrix A as follows: on the i-th column we put the vector ${\displaystyle f\left(e_i\right)}$ where ${\displaystyle E=(e_1,\cdots ,e_n)}$ is a basis of ${\displaystyle {\mathbb{R}}^n}$.

I have a question regarding matrix transformations. I am trying to transform a parallelogram into the unit square. I need to find a series of matrices which transforms each point.
For example, how do I find a, b, c and d (matrix) when (0,1) is part of the unit square and (3,10) is the original point?
[a b] [3 ] = [0] [c d] [10]

Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
$\left[\begin{array}{ccc}1& 3& -2\\ -4& h& 8\end{array}\right]$

Find the characteristic equation and the eigenvalues (and corresponding eigenvectors) of the matrix.
$$\left[\begin{array}{ccc}2& -2& 3\\ 0& 3& -2\\ 0& -1& 2\end{array}\right]$$
a) The characteristic equation
${\displaystyle {\lambda }^3-7{\lambda }^2+14\lambda -8=0}$
b) The eigenvalues
${\displaystyle ({\lambda }_1,{\lambda }_2,{\lambda }_3)=(1,2,4)}$
The corresponding eigenvectors
${\displaystyle x_1=-7,1,1?}$
${\displaystyle x_2=1,0,0}$
${\displaystyle x_3=13,-4,2?}$