Linear depiction is given $ T_{M2*2}(R)->R^2 $ $ T(A)=(a11+a22,a12+a21) $ Where A is a matrix \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}
$1$) Prove that $T$ is a linear depiction
$2$) Find the kernel of depiction $T$
$3$) Check if kerf $T$ is the total of antisymmetric tables $ 2*2 $
I managed to solve the first one but I have trouble solving number $2$) and $3$)!
The kernel of such transformation includes all matrices $M_{2\times 2}$ such that $T(M)=0$. This requires all matrices with $a_{11}+a_{22}=0$ and $a_{12}+a_{21}=0$ and leads to $a_{11}=-a_{22}$ and $a_{12}=-a_{21}$. So the dimension of the kernel is $2$. $$\ker(T)=\lbrace all \ 2\times 2 \ matrices \ with \ a_{11}=-a_{22} \ and \ a_{12}=-a_{21} \rbrace$$also the set of antisymmetric matrices is a special case of this one with $a_{11}=a_{22}=0$ and $a_{12}=-a_{21}$. So this subspace includes all of those kinds of matrices.