Let $V=\mathbb{R}^{2\times 2}$ be the vector space of $2\times 2$ matrices and let $L :V\to V$ be defined by $L(X) = \left[\begin{array}{cc} -4 &-2\cr -6 &-3 \end{array}\right] X$. Hint: The image of a spanning set is a spanning set for the image.
a. Find $L(\left[\begin{array}{cc} 3 &5\cr 0 &-3 \end{array}\right])=$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
b. Find a basis for $\ker (L )$:
{ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ , $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ }
c. Find a basis for $\text{ran} (L )$:
{ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ , $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ }