<s>[INST] null space of a matrix for a matrix with a 0 column I'm not a mathematician. Math is a hobby and currently I'm learning bits and pieces of linear algebra when I have time. Confused about a null space for the following matrix
\begin{bmatrix}
 0 & 0 \\
 0 & 2
\end{bmatrix} 
How does one find a null space for this matrix? As far as I understand I need to a such values so that the total would equal to $0$. So what I'm getting is
$$
N = \alpha \begin{bmatrix} 0 \\ 0\end{bmatrix} + 0 \begin{bmatrix} 0 \\ 2\end{bmatrix}
$$
Wouldn't a coefficient of $0$ make it a trivial solution?
 [/INST] We find the null space by inspection. (Systematic methods are demonstrated in the linked pages below)
Given 
$$
\mathbf{A} =
\left(
\begin{array}{cc}
 0 & 0 \\
 0 & 2 \\
\end{array}
\right)
$$
we note that
$$
\mathbf{A} 
\left(
\begin{array}{cc}
 1  \\
 0 \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
 0  \\
 0 \\
\end{array}
\right)
$$
Both $\color{red}{null}$ spaces are the same for your symmetric matrix:
$$
 \color{red}{\mathcal{N}\left( \mathbf{A} \right)} = \color{red}{\mathcal{N}\left( \mathbf{A}^{*} \right)}
=\text{span }
%
\left\{ \,
\color{red}{\left(
\begin{array}{cc}
 1  \\
 0 \\
\end{array}
\right)}
\, \right\}
%
$$

Consider the linear system
$$
\begin{align}
  \mathbf{A} x & = b \\
%
\left(
\begin{array}{cc}
 0 & 0 \\
 0 & 2 \\
\end{array}
\right)
%
\left(
\begin{array}{c}
 x_{1} \\
 x_{2} \\
\end{array}
\right)
%
&=
\left(
\begin{array}{c}
 b_{1} \\
 b_{2} \\
\end{array}
\right)
\end{align}
$$
and specify the data vector to classify the existence and uniqueness of solutions.

Existence and uniqueness
When the data vector has the form
$$
b = \left(
\begin{array}{c}
 0 \\
 b_{2} \\
\end{array}
\right)
$$
where $b_{2} \ne 0$, the solution exists and is unique. In fact, the solution is
$$
 x= 
\color{blue}{\left(
\begin{array}{c}
 0 \\
 \frac{1}{2} b_{2} \\
\end{array}
\right)}
$$
Existence without uniqueness
When the data vector has the form
$$
b = \left(
\begin{array}{c}
 b_{1} \\
 b_{2} \\
\end{array}
\right)
$$
where $b_{1} \ne 0$, and  $b_{2} \ne 0$, the solution exists and is not unique.
The least squares solution is 
$$
 x_{LS} =
\color{blue}{\left(
\begin{array}{c}
 0 \\
 \frac{1}{2} b_{2} \\
\end{array}
\right)} +
\alpha 
\color{red}{\left(
\begin{array}{c}
 1 \\
 0 \\
\end{array}
\right)}, \qquad \alpha\in \mathbb{C}
$$
No existence
When the data vector has the form
$$
b = \color{red}{\left(
\begin{array}{c}
 b_{1} \\
 0 \\
\end{array}
\right)}
$$
where $b_{1} \ne 0$, the solution does not exist.

Read more on MSE

Formal steps for computing null spaces:
Deriving left nullspace of matrix from EA=R,
Find base and dimension of given subspace
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