How to show that transformation L is lineair?

Question

Define L : $\mathbb{R}$[X, Y]$_\leq$ $_2$ $\longrightarrow$ $\mathbb{R}$[X]$_\leq$ $_2$

By L : f(X, Y) $\longrightarrow$ f(X, 2)

How to show that transformation L is lineair?

Of course I first tried it myself, but I ended up with a non-linear transformation and that cannot be correct because L is linear. I am very curious if someone can elaborate this question so that I can see what I did wrong. Thanks in advance for your help!

Answer 1

We want to show that for all $f,g \in \Bbb R_{\leq 2}[X,Y]$ and $k \in \Bbb R$, $L(f + k g) = L(f) + kL(g)$. To that end, note that \begin{align} L(f + kg) & = L(f(x,y) + kg(x,y)) = f(x,2) + kg(x,2)\\ L(f) + kL(g) &= [f(x,2)] + k[g(x,2)] = f(x,2) + kg(x,2). \end{align} The result is indeed the same, so $L$ is linear.

Answer 2

Hint: This transformation can be written as

$L: aX^2 + b XY + c Y^2 + d X + e Y + f \mapsto aX^2 + 2bX + 4c + dX+ 2e + f= aX^2 + (2b+d)X + (4c+2e+f).$