Matrix of a Linear mapping, Null Space ,Range Space  explain with examples and proper explaination

### 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

Understand the concepts of matrix representation of a linear mapping, null space (kernel), and range space (image) with detailed examples and explanations. Learn how to find the matrix representation of a linear transformation, determine the null space of a matrix, and identify the range space of a linear transformation.

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