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. FindL(\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. }

a. Find

b. Find a basis for

{

c. Find a basis for

{

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