Is the Map $T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3$ a Linear Transformation?

Problem 678

Let $C ([0, 3] )$ be the vector space of real functions on the interval $[0, 3]$. Let $\mathrm{P}_3$ denote the set of real polynomials of degree $3$ or less.

Define the map $T : C ([0, 3] ) \rightarrow \mathrm{P}_3 $ by
\[T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3.\]

Determine if $T$ is a linear transformation. If it is, determine its nullspace.

We will show that $T$ satisfies both of the axioms for linear transformations. Suppose $f, g \in C([0, 3])$. Then
\begin{align*}
&T(f+g)(x)\\
&= (f+g)(0) + (f+g)(1) \cdot x + (f+g)(2) \cdot x^2 + (f+g)(3) \cdot x^3 \\
&= f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3 + g(0) + g(1) \cdot x + g(2) \cdot x^2 + g(3) \cdot x^3 \\
&= T(f)(x) + T(g)(x) . \end{align*}

Thus we see that $T(f+g) = T(f) + T(g)$.

Next, suppose $f \in C([0, 3])$ and $c \in \mathbb{R}$. Then,
\begin{align*}
&T(cf)(x)\\
&= (cf)(0) + (cf)(1) \cdot x + (cf)(2) \cdot x^2 + (cf)(3) \cdot x^3 \\
&= c f(0) + c f(1) \cdot x + c f(2) \cdot x^2 + c f(3) \cdot x^3 \\
&= c ( f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3 ) \\
&= c T(f)(x) . \end{align*}

This shows that $T$ is a linear transformation.

The nullspace of $T$

Next, we will find the nullspace of $T$. Suppose that $f \in C([0, 3])$ such that $T(f) = 0$. That means that
\[f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3 = 0.\]

This implies that $f(0) = f(1) = f(2) = f(3) = 0$. On the other hand, if $f(0)=f(1)=f(2)=f(3)=0$, then $T(f)(x) = 0$. Thus, the nullspace is
\[\mathcal{N}(T) = \{ f \in C([0, 3]) \mid f(0) = f(1) = f(2) = f(3) = 0 \}.\]

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