Answer
See below
Work Step by Step
Given $A=\begin{bmatrix}
2 & -4\\ 1 & 2 \\ -3 & -5
\end{bmatrix}$
Since $x,y \in nullspace (A)$ we obtain
$Ax=0\\
\begin{bmatrix}
2 & -4 \\ 1 & 2 \\ -3 & -5
\end{bmatrix}\begin{bmatrix}
x \\ y
\end{bmatrix}=0\\
\begin{bmatrix}
2x -4y \\ x +2y \\ -3x -5y
\end{bmatrix}=0\\
\rightarrow 2x-4y=0\\
x+2y=0\\
-3x-5y=0\\
\rightarrow x=-2y\\
-3x-5y=0\\
\rightarrow 3(-2y)-5y=0\\
\rightarrow y=0\\
x=0$
Hence, nullspace $(A) =\{(0,0)\}$
