## 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.

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