I found [T]E to be:
2 2 4 8
0 0 -4 -12
0 0 2 6
0 0 0 0
but how can I find the kernel kerT? Appreciate any help!
Then use that definition!
$\displaystyle T_E(v)= \begin{bmatrix} 2 & 2 & 4 & 8 \\ 0 & 0 & -4 & -12 \\ 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}u \\ x\\ y \\ z\end{bmatrix}$$\displaystyle = \begin{bmatrix}2u+ 2x+ 4y+ 8z \\-4 y+ 12 z\\ 2y+ 2z \\ 0 \end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0\end{bmatrix} $
which gives the three equations 2u+ 2x+ 4y+ 8z= 0, -4y- 12z= 0, and 2y+ 6z= 0. What values of u, x, y, and z satisfy those equations?