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.
Best Answer
Suppose instead of $\mathbb R$ we had used the field $\mathbb Z_2 = \mathbb Z/(2\mathbb Z),$ in other words, the set $\{0,1\}$ with the usual operations $+$ and $\cdot$ modulo $2.$
Now we would be asking about $T: \mathbb Z_2^2 \to \mathbb Z_2^3,$ with $T\left( x, y \right) = \left( y, x, x^2 + y^2 \right).$
Every part of your proof would then work just as well as it did for $T: \mathbb R^2 \to \mathbb R^3,$ with the exception of the conclusion. The conclusion would be false, because $x^2 + y^2 = x + y$ when $x, y \in \mathbb Z_2.$
I do not see any point in the proof where you invoke any property of $\mathbb R$ that $\mathbb Z_2$ does not have. Therefore I would say the proof is not valid.
In order to make a valid proof, you could invoke (for example) the fact that $\mathbb R$ contains an element named $2$ that is distinct from $0$ and $1,$ and you could have used the properties of that element to find a counterexample to the statement $x^2 + y^2 = x + y$ for $x,y \in \mathbb R.$
I think it is noteworthy that you used many more facts than you needed in this proof, which I think also is a bad thing to do in a proof, but of course that alone does not invalidate a proof.