Answer
a basis for the nullspace of $A$ consists of the vector
$$ \left[\begin{aligned} 0\\ 0 \end{aligned}\right].$$
Work Step by Step
Given the matrix
$$
A=\left[\begin{array}{ll}{2} & {-1} \\ {1} & {3}\end{array}\right].
$$
The reduced row echelon form is given by
$$
\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]
.
$$
The corresponding system is
$$
\begin{aligned} x_{1} &=0 \\
x_2&=0 \end{aligned}.
$$
The solution of the above system is $x_1=0$ and $x_2=0$. This means that the solution space of $Ax = 0 $ consists of the zero vector,
following form
$$x= \left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]= \left[\begin{aligned} 0\\ 0 \end{aligned}\right].$$
So, a basis for the nullspace of $A$ consists of the vector
$$ \left[\begin{aligned} 0\\ 0 \end{aligned}\right].$$
