M22 → R Matrix Transformation Kernel

Question

For a transformation such as this, how does one determine the form of the kernel? Is it simply making the right side equal to zero, solving for each individual variable, and then creating a matrix with these 'new' a,b,c,d values?

Answer 1

Thats right, finding the kernel of this linear transformation is equivalent to solving a linear equation which can be written in the following form.

$\begin{pmatrix}3 & -4 & 1 & -1 \end{pmatrix}\begin{pmatrix}a \\ b \\ c \\ d\end{pmatrix} = 0$

$\begin{pmatrix}3 & -4 & 1 & -1 \end{pmatrix} \sim \begin{pmatrix}1 & -4/3 & 1/3 & -1/3 \end{pmatrix}$

And thus the vector of $(a,b,c,d)$ is an element of

$\left<\begin{pmatrix}4/3 \\ 1 \\ 0 \\ 0\end{pmatrix},\begin{pmatrix}-1/3 \\ 0 \\ 1 \\ 0\end{pmatrix},\begin{pmatrix}1/3 \\ 0 \\ 0 \\ 1\end{pmatrix} \right>$

Using the $\mathbb{R}^4 \rightarrow \mathbb{R}^{2\times2}$ isomorphism you can translate this into the solution for your initial problem.