Question

let L be a linear transformation defined by (see inside)
Determine the dimention of kernel of L.

Answer 1

\[L( \left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right]) =\left[\begin{matrix}0 & -3 \\ -2 & -4\end{matrix}\right]\]
\[L(\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right])=\left[\begin{matrix}-2 & -3 \\ 0 & 1\end{matrix}\right]\]
\[L(\left[\begin{matrix}0 & 0 \\ 1 & 0\end{matrix}\right])=\left[\begin{matrix}0 & 1 \\ 4 & -4\end{matrix}\right]\]
\[L(\left[\begin{matrix}0 & 0 \\ 0 & 1\end{matrix}\right])=\left[\begin{matrix}0 & -2 \\ 4 & -4\end{matrix}\right]\]

Answer 2

\[T(\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]) = \left[\begin{matrix}-2b & -3a -3b +d -2d \\ -2a+4c +4d & -4a+b-4c-4d\end{matrix}\right]\]

Answer 3

Sorry top right should be: -3a - 3b + c -2d
T(kernel) = 0

so solve this system:

-2b=0
-3a - 3b + c -2d=0
-2a+4c-4d=0
-4a+b-4c-4d=0

Answer 4

thnks with that i was able to do get to the answer, so nullity = 0 and rank =4
thanks