Taking the Third Order Taylor Polynomial is a Linear Transformation
Problem 675
The space $C^{\infty} (\mathbb{R})$ is the vector space of real functions which are infinitely differentiable. Let $T : C^{\infty} (\mathbb{R}) \rightarrow \mathrm{P}_3$ be the map which takes $f \in C^{\infty}(\mathbb{R})$ to its third order Taylor polynomial, specifically defined by
\[ T(f)(x) = f(0) + f'(0) x + \frac{f^{\prime\prime}(0)}{2} x^2 + \frac{f^{\prime \prime \prime}(0)}{6} x^3.\]
Here, $f’, f^{\prime\prime}$ and $f^{\prime \prime \prime}$ denote the first, second, and third derivatives of $f$, respectively.
To prove that $T$ is a linear transformation, we must check that it satisfies the two axioms for linear transformations. First, suppose $f, g \in C^{\infty} ( \mathbb{R} )$. Then basic properties of the derivative tell us that
\begin{align*}
(f + g)'(x) &= f'(x) + g'(x) \\
(f+g)^{\prime\prime}(x) &= f^{\prime\prime}(x) + g^{\prime\prime}(x)\\
(f+g)^{\prime \prime \prime}(x) &= f^{\prime \prime \prime}(x) + g^{\prime \prime \prime}(x).
\end{align*}
And so we can see
\begin{align*}
&T(f+g)(x)\\
&= (f+g)(0) + (f+g)'(0) x + \frac{(f+g)^{\prime\prime}(0)}{2} x^2 + \frac{(f+g)^{\prime \prime \prime}(0)}{6} x^3 \\[6pt]
&= f(0) + g(0) + f'(0) x + g'(0) x + \frac{ f^{\prime\prime}(0) }{2} x^2 + \frac{g^{\prime\prime}(0)}{2} x^2 + \frac{f^{\prime \prime \prime}(0)}{6} x^3 + \frac{g^{\prime \prime \prime}(0)}{6} x^3 \\[6pt]
&= f(0) + f'(0) x + \frac{ f^{\prime\prime}(0) }{2} x^2 + \frac{f^{\prime \prime \prime}(0)}{6} x^3 + g(0) + g'(0) x + \frac{g^{\prime\prime}(0)}{2} x^2 + \frac{g^{\prime \prime \prime}(0)}{6} x^3 \\[6pt]
&= T(f)(x) + T(g)(x) .
\end{align*}
Now for a scalar $c \in \mathbb{R}$, the equality $(cf)'(x) = c f'(x)$ is a basic property of the derivative. Then,
\begin{align*}
T( cf )(x) &= c f(0) + (cf)'(0) x + \frac{(cf)^{\prime\prime}(0)}{2} x^2 + \frac{(cf)^{\prime \prime \prime}(0)}{6} x^3 \\
&= c f(0) + c f'(0) x + c \frac{ f^{\prime\prime}(0) }{2} x^2 + c \frac{ f^{\prime \prime \prime}(0) }{6} x^3 \\
&= c \left( f(0) + f'(0) x + \frac{ f^{\prime\prime}(0) }{2} x^2 + \frac{ f^{\prime \prime \prime}(0) }{6} x^3 \right) \\
&= c T(f)(x)
\end{align*}
Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation
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 of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$
Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in […]
Differentiation is a Linear Transformation
Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\]
for any $f(x)\in […]
Differentiating Linear Transformation is Nilpotent
Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]
(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials
Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by
\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = […]
Matrix Representations for Linear Transformations of the Vector Space of Polynomials
Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.
Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.
For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]