Matrix of a Linear Mapping

A linear mapping (or linear transformation) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. Suppose we have two vector spaces $V$ and $W$, and a linear mapping $T: V \rightarrow W$. If $\{v_1, v_2, \dots, v_n\}$ is a basis for $V$, and $\{w_1, w_2, \dots, w_m\}$ is a basis for $W$, then the matrix of the linear transformation $T$ relative to these bases is a matrix $A$ such that:

$T(v_i) = \sum_{j=1}^m A_{ji} w_j \quad \text{for } i = 1, 2, \dots, n$

Here, $A_{ji}$ are the elements of the matrix $A$.

Example: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation defined by $T(x, y) = (2x + y, x - y)$. To find the matrix representation of $T$, we apply $T$ to the standard basis vectors $(1, 0)$ and $(0, 1)$:

$T(1, 0) = (2 \cdot 1 + 0, 1 - 0) = (2, 1)$ $T(0, 1) = (2 \cdot 0 + 1, 0 - 1) = (1, -1)$

So, the matrix of $T$ is:

$A = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix}$

Null Space (Kernel)

The null space (or kernel) of a linear mapping $T: V \rightarrow W$ is the set of all vectors in $V$ that are mapped to the zero vector in $W$. Mathematically, the null space $N(T)$ is defined as:

$N(T) = \{v \in V : T(v) = 0 \}$

Example: Consider the matrix $A$ from the previous example:

$A = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix}$

To find the null space, solve $A \mathbf{x} = 0$:

$\begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

This system corresponds to:

$2x_1 + x_2 = 0 \quad \text{and} \quad x_1 - x_2 = 0$

From the second equation, $x_1 = x_2$. Substituting into the first equation, $2x_1 + x_1 = 0$, so $3x_1 = 0$, hence $x_1 = 0$ and $x_2 = 0$. Therefore, the null space consists only of the zero vector:

$N(T) = \{(0, 0)\}$

Range Space (Image)

The range space (or image) of a linear mapping $T: V \rightarrow W$ is the set of all vectors in $W$ that can be expressed as $T(v)$ for some $v \in V$. The range space $R(T)$ is defined as:

$R(T) = \{ T(v) : v \in V \}$

Example: Using the matrix $A$ from earlier, the range space is the span of the column vectors of $A$. The columns of $A$ are $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$. Thus:

$R(T) = \text{span} \left\{ \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \end{pmatrix} \right\}$

To check if these vectors span the whole space $\mathbb{R}^2$, we examine the determinant of the matrix formed by these vectors:

[ \text{det} \begin{pmatrix} 2