Finding a basis

Question

Finding the basis for the kernel of:

\begin{pmatrix} a & b \\c & d\end{pmatrix}

$which$ $maps$ $to:$

\begin{pmatrix} a \\a\\3a + b \end{pmatrix}

It's all complex, but I'm not sure if that's relevant!

Answer 1

So you have the map $F:\mathbb C^{2\times 2} \longrightarrow \mathbb C^3, \ \begin{pmatrix}a&b\\c&d \end{pmatrix} \mapsto\begin{pmatrix}a\\a\\3a+b \end{pmatrix}$ and you want to find $\ker F$.
By definition $$\ker F=\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:F\left(\begin{pmatrix}a&b\\c&d \end{pmatrix}\right)=\begin{pmatrix}0\\0\\0 \end{pmatrix}\right\}\\ =\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:\begin{pmatrix}a\\a\\3a+b \end{pmatrix}=\begin{pmatrix}0\\0\\0 \end{pmatrix}\right\}\\ =\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:a=0,3a+b=0\right\}=\cdots$$ After that find the basis.

An answer is

$\left\{\begin{pmatrix}0&0\\1&0 \end{pmatrix},\begin{pmatrix}0&0\\0&1 \end{pmatrix}\right\}$