Let $F\left(\begin{matrix} a & b \\ c & d \end{matrix}\right) =\left(\begin{matrix} a & 0 \\ 0 & d \end{matrix}\right)$ be a linear function with $F : M(2\times2) \rightarrow M(2\times2)$.
- Which of the following are kernels and images of $F$? $ a. \left(\begin{matrix} 1 & 2 \\ -1 & 3 \end{matrix}\right) \\ b. \left(\begin{matrix} 0 & 4 \\ 2 & 0 \end{matrix}\right) \\ c. \left(\begin{matrix} 3 & 0 \\ 0 & -3 \end{matrix}\right) $
- Describe kernel and image by specifying a base (for each)
My answer and thoughts
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- b is a kernel, because running it through $F$ yelds $0$.
- 1. a and 1. c are images (but I'm not sure in case of 1.c, does this have to hold $b\neq c$? In case of 1.c they both are $0$)
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- I don't know how to write it out formally correct, but the kernel has to look like this $\left(\begin{matrix} 0 & x \\ y & 0 \end{matrix}\right)$ for any $x,y$.
- I have no idea about how to write an image as a base.
For the kernel, I suspect something like this must hold
$ \lambda_1\left(\begin{matrix} 0 & x \\ y & 0 \end{matrix}\right)+ \lambda_2\left(\begin{matrix} 0 & x \\ y & 0 \end{matrix}\right)= \left(\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}\right) $ which would mean that $x, y \neq 0$ for them to be linear independent.