I have the following transformation $T:\mathbb{R}^2 \longrightarrow \mathbb{R}^3$ defined by $T\left( x, y \right) = \left( y, x, x^2 + y^2 \right).$ I know the transformation is not linear but would like to prove it, so I deviced the following "proof."
We know every linear transformation $T$ has a unique matrix representation for the standard basis of $\mathbb{R}^2,$ which is given by $$A = \left[ \begin{array}{ccc} T(\mathbf{e}_1) & T(\mathbf{e}_2) \\\end{array} \right],$$ and this matrix $A$ would move me back to the linear transformation by $T\left( \mathbf{x} \right) = A \mathbf{x}.$
So, I assume $T$ is a linear transformation and construct it standard matrix representation, which would be $$A = \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{array} \right).$$
Now, to get my original transfomation back I would have to do $$T\left( \mathbf{x} \right) = A \mathbf{x} = \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{array} \right) \cdot \left( \begin{array}{ccc} x \\ y \end{array} \right) = \left( \begin{array}{ccc} y \\ x \\ x+y \end{array} \right).$$
Since this transformation I got is not the original one, I conclude $T$ is not a linear transformation.
My question is, the above reasoning is correct?
And in general, can I apply this method to prove or disprove any transformation is a linear transformation?
EDIT:
Please do not sugegst alternative methods of proof; I know them well. All I need is to know if the method described works.