Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions?

Linear Transformation problems and solutions

Problem 677

Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$.

Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$.

 
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Solution.

We claim that $T$ is not a linear transformation.

We can see this by seeing
\[ T(f+g)(x) = ( f(x) + g(x) )^2 = f^2(x) + 2 f(x) g(x) + g^2 (x) \] while
\[ T(f)(x) + T(g)(x) = f^2(x) + g^2(x) . \] Because $T(f+g) \neq T(f) + T(g)$, we see that $T$ is not a linear transformation.


For a specific example, consider $f(x) = g(x) = x$. Then
\[ T( f+g)(x) = (2x)^2 = 4x^2 , \] while
\[ T(f)(x) + T(g)(x) = x^2 + x^2 = 2x^2 . \] Clearly $T(f+g) \neq T(f) + T(g)$.


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