The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$

Problem 673

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 $f \in \mathrm{P}_3$,
\[T (f) (x) = ( x^2 – 2) f(x).\]

Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_3$ and $\mathrm{P}_{5}$.

We must check that $T$ satisfies the two axioms for linear transformations. Suppose $f, g \in \mathrm{P}_3$. Then
\[T(f+g)(x) = (x^2-2) (f+g)(x) = (x^2 – 2) f(x) + (x^2 – 2) g(x) = T(f)(x) + T(g)(x).\]

Second, if $c \in \mathbb{R}$ then
\[T( cf )(x) = (x^2 – 2) ( cf) (x) = c (x^2 – 2) f(x) = c T(f)(x).\]

The two axioms are satisfied, and so $T$ is a linear transformation.

Now we find its matrix representation relative to the standard basis $B = \{ 1 , x , x^2 , x^3 \}$ of $\mathrm{P}_3$ and the standard basis $C = \{ 1 , x , x^2 , x^3 , x^4 , x^5 \} $ . To do this, for every $f \in B$ we calculate the coordinate vector of $T(f)$ relative to the basis $C$.

For example, we find that $T(1) = x^2 – 2$. The coordinate vector for this element, relative to $C$, is
\[ [ x^2 – 2 ]_{C} = \begin{bmatrix} -2 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}.\]
Similarly, we find that $T(x) = x^3 – 2x$. The coordinate vector for this element, relative to $C$,
\[ [ x^3 – 2x ]_{C} = \begin{bmatrix} 0 \\ -2 \\ 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}.\]

Performing this process for the rest of the elements of $B$, we get
\[ [T(x^2)]_{C} = [ x^4 – 2x^2 ]_{C} = \begin{bmatrix} 0 \\ 0 \\ -2 \\ 0 \\ 1 \\ 0 \end{bmatrix} \quad [T(x^3)]_{C} = [ x^5 – 2x^3 ]_{C} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ -2 \\ 0 \\ 1 \end{bmatrix}.\]
And now putting these four column vectors in order, we get the matrix representation for $T$:
\[ [T]_{B}^{C} = \begin{bmatrix} -2 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 1 & 0 & -2 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.\]

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 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) = […]

Null Space, Nullity, Range, Rank of a Projection Linear Transformation
Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) […]

Subspace Spanned By Cosine and Sine Functions
Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […]

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 […]

The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.
Then find the matrix representation of the linear transformation $T$ with respect to the […]

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 […]

in case of T(x),

You have one extra row.

instead of 6 there are 7.

one 0 is extra.

Dear Himanshu Yadav,

Thank you for catching the typo. I modified it accordingly. Thanks!