Let $U$ and $V$ be vector spaces over a scalar field $\F$.
Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.

(a) Prove that $T:U\to V$ is a linear transformation.
(Hence, $T$ is called the zero transformation.)

(b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.

(a) Prove that $T:U\to V$ is a linear transformation.

Let $\mathbf{u}_1, \mathbf{u}_2\in U$ and $r$ be a scalar, that is, $r\in \F$.
It follows from the definition of $T$ that
\begin{align*}
T(\mathbf{u}_1)&=\mathbf{0}_V, \quad T(\mathbf{v}_2)=\mathbf{0}_V, \\
T(\mathbf{u}_1+\mathbf{u}_2)&=\mathbf{0}_V, \quad T(r\mathbf{u}_1)=\mathbf{0}_V
\end{align*}
since $\mathbf{u}_1+\mathbf{u}_2, r\mathbf{u}_1\in U$.
Hence we have
\begin{align*}
T(\mathbf{u}_1+\mathbf{u}_2)&=\mathbf{0}_V=\mathbf{0}_V+\mathbf{0}_V=T(\mathbf{u}_1)+T(\mathbf{u}_2)\\
T(r\mathbf{u}_1)&=\mathbf{0}_V=r\mathbf{0}_V=rT(\mathbf{u}_1).
\end{align*}
Since these equalities holds for all $\mathbf{u}_1, \mathbf{u}_2\in U$, and $r\in \F$, the map $T:U\to V$ is a linear transformation.

(b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.

The null space $\calN(T)$ of $T$ is, by definition,
\begin{align*}
\calN(T)=\{\mathbf{u}\in U \mid T(\mathbf{u})=\mathbf{0}_V\}.
\end{align*}
Since $T(\mathbf{u})=\mathbf{0}_V$ for every $\mathbf{u}\in U$, we obtain
\[\calN(T)=U.\]

The range $\calR(T)$ of $T$ is, by definition,
\[\calR(T)=\{\mathbf{v} \in V \mid \text{there exists } \mathbf{u}\in U \text{ such that } T(\mathbf{u})=\mathbf{v}\}.\]

Since every vector of $U$ is mapped into $\mathbf{0}_V$, we have
\[\calR(T)=\{\mathbf{0}_V\}.\]

A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$
Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.
Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).
Hints.
You may use the folowing facts.
A linear […]

Linear Properties of Matrix Multiplication and the Null Space of a Matrix
Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.
Then find $A\mathbf{w}$.
Hint.
Recall that the null space of an […]

Dimension of Null Spaces of Similar Matrices are the Same
Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]

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 […]

Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation
Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\]
for each $X\in V$.
(a) Prove that $T:V\to V$ is a linear transformation.
(b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]

Find a Value of a Linear Transformation From $\R^2$ to $\R^3$
Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}$ are unit vectors of $\R^2$ and […]

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 kernel of $T$ is $n-1$.
(The kernel of $T$ is also called the null space of $T$.)
(b) Let […]

Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$
Let $V$ be a vector space over the field of real numbers $\R$.
Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.
Proof.
Since $V$ is an $n$-dimensional vector space, it has a basis
\[B=\{\mathbf{v}_1, \dots, […]