Answer
(a) The nullspace of $Ax = 0 $ consists of the vectors on the following form
$$x= \left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]= \left[\begin{aligned}\frac{8}{5}t\\t \end{aligned}\right]=t \left[\begin{aligned}\frac{8}{5}\\1 \end{aligned}\right] .$$
(b) The nullity is $1$.
(c) The rank of $A$ is $1$.
Since $A$ has two columns, one can see that
$$\text{rank}(A)+\text{nullity}(A)=1+1=2.$$
Work Step by Step
Given the matrix
$$
A=\left[ \begin {array}{cc} 5&-8\\ -10&16\end {array}
\right]
$$
The reduced row echelon form is
$$
\left[ \begin {array}{cc} 1&-\frac{8}{5}\\ 0&0\end {array}
\right]
.
$$
The corresponding system is
$$
\begin{aligned} x_{1}-\frac{8}{5}x_{2} &=0\\
\end{aligned}.
$$
The solution of the above system is $x_1= \frac{8}{5}t$,$x_2=t$.
(a) The nullspace of $Ax = 0 $ consists of the vectors on the following form
$$x= \left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]= \left[\begin{aligned}\frac{8}{5}t\\t \end{aligned}\right]=t \left[\begin{aligned}\frac{8}{5}\\1 \end{aligned}\right] .$$
(b) The nullity is $1$.
(c) The rank of $A$ is $1$.
Since $A$ has two columns, one can see that
$$\text{rank}(A)+\text{nullity}(A)=1+1=2.$$
