Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials?

Problem 674

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 $T : \mathrm{P}_4 \rightarrow \mathrm{P}_{4}$ be the map defined by, for $f \in \mathrm{P}_4$,
\[ T (f) (x) = f(x) – x – 1.\]

Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_4$.

To see this, consider $f(x) = 1 + x$. Then
\[T(f)(x) = 1 + x – x – 1 = 0.\]
On the other hand,
\[T(1) + T(x) = (1 – x – 1) + (x – x – 1) = -x – 1.\]
Because $T(1+x) \neq T(1) + T(x)$, we see that $T$ is not a linear transformation.

Solution 2.

Recall the fact that any linear transformation maps the zero vector to the zero vector.
The zero vector in $\mathrm{P}_n$ is the zero polynomial $\theta(x)=0$.

Then we have
\[T(\theta)(x)=\theta(x)-x-1=-x-1 \neq \theta(x).\]

Thus, the map $T$ does not send the zero vector $\theta(x)$ to the zero vector $\theta(x)$.
Hence, $T$ is not a linear transformation.

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 […]

Taking the Third Order Taylor Polynomial is a Linear Transformation
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 […]

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) = […]

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 […]

Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation?
Determine whether the function $T:\R^2 \to \R^3$ defined by
\[T\left(\, \begin{bmatrix}
x \\
y
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_+y \\
x+1 \\
3y
\end{bmatrix}\]
is a linear transformation.
Solution.
The […]

A Linear Transformation Maps the Zero Vector to the Zero Vector
Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.
(The Ohio State University Linear Algebra […]