Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation

Problem 679

For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by
\[ T(f)(x) = f(x^2).\]

Determine if $T$ is a linear transformation.

If it is, find the matrix representation for $T$ relative to the basis $\mathcal{B} = \{ 1 , x , x^2 \}$ of $\mathrm{P}_2$ and $\mathcal{C} = \{ 1 , x , x^2 , x^3 , x^4 \}$ of $\mathrm{P}_4$.

To prove that $T$ is a linear transformation, we must show that it satisfies both axioms for linear transformations. For $f, g \in \mathrm{P}_2$, we have
\[T( f+g )(x) = (f+g)(x^2) = f(x^2) + g(x^2) = T(f)(x) + T(g)(x)\]
while for a scalar $c \in \mathbb{R}$, we have
\[ T( c f )(x) = (cf)(x^2) = c f(x^2) = c T(f)(x).\]
We see that $T$ is a linear transformation.

The matrix representation for $T$

To find its matrix representation, we must calculate $T(f)$ for each $f \in \mathcal{B}$ and find its coordinate vector relative to the basis $\mathcal{C}$. We calculate
\[T(1) = 1 , \quad T(x) = x^2 , \quad T(x^2) = x^4.\]
Each of these is an element of $\mathcal{C}$. Their coordinate vectors relative to $\mathcal{C}$ are thus
\[[ T(1) ]_{\mathcal{B}} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} , \quad [ T(x) ]_{\mathcal{B}} = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} , \quad [ T(x^2) ]_{\mathcal{C}} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}.\]

The matrix representation of $T$ is found by combining these columns, in order, into one matrix:

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