Find basis for the image and the kernel of a linear map How do I calculate a basis for $\ker (\varphi)$ and $\operatorname{im}(\varphi)$?
Where, given 
$$ A = \begin{pmatrix} 
1 & -1\\
-1 & 1 \\
\end{pmatrix}$$
we define
$$ \begin{matrix}
\varphi: \mathbb{R}^{2 \times 2} \to \mathbb{R}^{2 \times 2} \\
X \mapsto XA+A^t X^t
\end{matrix}$$
Let $B$ be the standard basis for $\mathbb{R}^{2 \times 2}$ :
$$B =\left\{ \begin{pmatrix} 
1 & 0\\
0 & 0 
\end{pmatrix}, \begin{pmatrix} 
0 & 1\\
0 & 0\\ 
\end{pmatrix},\begin{pmatrix} 
0 & 0\\
1 & 0\\ 
\end{pmatrix},\begin{pmatrix} 
0 & 0\\
0 & 1\\ 
\end{pmatrix} \right\}$$
Calculate $\textsf{M}_B(\varphi)$ we come to $$\textsf{M}_B(\varphi) = \begin{pmatrix} 
0 & 0 & 0 & 0\\
-1 & 1 & -1 & 1 \\
1 & -1 & 1 & -1\\
0 & 0 & 0 & 0\\
\end{pmatrix}$$
We calculate a basis for the kernel like this:
If 
$$X:= \begin{pmatrix} 
a & b\\
c & d\\ 
\end{pmatrix}$$
then $$\varphi(X) = \begin{pmatrix} 
a-b & -a+b\\
c-d & -c+d\\ 
\end{pmatrix}+\begin{pmatrix} 
a-b & c-d\\
-a+b & -c+d\\ 
\end{pmatrix} = \begin{pmatrix} 
2a-2b & -a+b+c-d\\
c-d-a+b & -2c+2d\\ 
\end{pmatrix}$$
Now we have to look, for what values $$\begin{pmatrix} 
2a-2b & -a+b+c-d\\
c-d-a+b & -2c+2d\\ 
\end{pmatrix}$$
by definition, the kernel of a linear transformation is $\varphi(X) = 0$, therefore our basis for $\ker(\varphi)$ should be 
$$\left\{ \begin{pmatrix} 1 & 1 \\ 0 & 0  \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 1  \end{pmatrix} \right\}$$
Now here comes the part where I'm confused. How do I calculate a basis for $\operatorname{im}(\varphi)$ ?
$\textsf{M}_B(\varphi)$ is the transformation matrix. I've read that you'd just transpose the matrix $\textsf{M}_B(\varphi)$ and row reduce to calculate a basis. I just don't get it.
The solution according to the solution it should be the basis $$ \left\{\begin{pmatrix}
0 & 1 \\
1 & -2  \end{pmatrix}, \begin{pmatrix}
1 & 0 \\
0 & -1  \end{pmatrix} \right\}$$
Also little side questions. I do know about 
$$\dim(A) = \dim(\operatorname{im} A) + \dim(\ker A)$$
but how exactly do you know the dimension of the Kernel/Image?

We can represent a $2\times 2$ matrix as a $1\times 4$ vector.
i.e. $\pmatrix{c_1\\c_2\\c_3\\c_4} = c_1\pmatrix{1&0\\0&0}+ c_2\pmatrix{0&1\\0&0}+c_3\pmatrix{0&0\\1&0}+c_4\pmatrix{0&0\\0&1}$
$\phi(X) = \pmatrix{2&-2&0&0\\-1&1&1&-1\\-1&1&1&-1\\0&0&2&-2}$
The linearly independent columns of the matrix above will give the image of the transformation.
