For an integer $n > 0$, 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 $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,
\[T (f) (x) = x f(x).\]

Prove that $T$ is a linear transformation, and find its range and nullspace.

We must show that for polynomials $f, g \in \mathrm{P}_n$ and scalar $c \in \mathbb{R}$, the function $T$ satisfies $T(f+g) = T(f) + T(g)$ and $T(cf) = c T(f)$. The first is checked using associativity of multiplication:
\[T(f+g)(x) = x (f+g)(x) = x f(x) + x g(x) = T(f)(x) + T(g)(x) . \]
Similarly,
\[T(cf)(x) = x (cf)(x) = c x f(x) = c T(f)(x).\]

The nullspace of $T$

The nullspace of $T$ is the set of polynomials $f(x)$ such that $T(f) = 0$. That is, $x f(x) = 0$. But this product can only be $0$ if one of the terms being multiplied is $0$. Because $x \neq 0$, we must have that $f(x) = 0$. Thus the nullspace is the trivial vector subspace $\{ 0 \}$.

The range of $T$

The range of $T$ is the set of polynomials of the form $x f(x)$ for an arbitrary polynomial $f \in \mathrm{P}_n$. We can calculate this by calculating $T$ for a basis of $\mathrm{P}_n$. Let $B = \{ 1 , x , x^2 , \cdots , x^n \}$ be a basis of $\mathrm{P}_n$.

We see that $T(x^i) = x^{i+1}$ for $ 0 \leq i \leq n$, and so the image of the basis is the set
\[T( B ) = \{ x , x^2 , x^3 , \cdots , x^{n+1} \}.\]
Then the range of $T$ must be the span of this set. Specifically,
\[\mathcal{R} ( T ) = \Span ( T( B ) ) = \Span ( x , x^2 , x^3 , \cdots , x^{n+1} ).\]

Linear Transformation to 1-Dimensional Vector Space and Its Kernel
Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.
(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.
(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]

Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$
Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
4
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
2 \\
5
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6 […]

A Matrix Representation of a Linear Transformation and Related Subspaces
Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\]
(a) Find a matrix $A$ such that […]

Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known
Let $T: \R^2 \to \R^2$ be a linear transformation.
Let
\[
\mathbf{u}=\begin{bmatrix}
1 \\
2
\end{bmatrix}, \mathbf{v}=\begin{bmatrix}
3 \\
5
\end{bmatrix}\]
be 2-dimensional vectors.
Suppose that
\begin{align*}
T(\mathbf{u})&=T\left( \begin{bmatrix}
1 \\
[…]

Find a Basis for the Range of a Linear Transformation of Vector Spaces of Matrices
Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by
\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b - d & -3c \\ 2b - c […]

Find an Orthonormal Basis of the Range of a Linear Transformation
Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\]
Find an orthonormal basis of the range of $T$.
(The Ohio […]

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 […]