Introduction

1. Understanding Linear Mappings Between Vector Spaces

Definition

A linear transformation is a mapping between two vector spaces that preserves vector addition and scalar multiplication. In simple terms, linear transformations ensure that the structure of a vector space is maintained during the mapping.

Mathematically, a function $T:V→W$ between two vector spaces $V$ and $W$ (over the same field, such as real numbers $\mathbb{R}$) that satisfies two main properties::

1. Additivity (preserves vector addition): $T(u + v) = T(u) + T(v), \quad \forall u, v \in V$.\

2. Homogeneity (preserves scalar multiplication): $T(c u) = c T(u), \quad \forall c \in \mathbb{R}, u \in V.$

These two properties ensure that a linear transformation maintains the "linear structure" of a vector space, such as straight lines, scalar multiples, and sums.

Matrix Representation of Linear Transformations

Any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be represented as a matrix $A \in \mathbb{R}^{m \times n}$:

$T(x) = Ax,$

where $\mathbf{x} \in \mathbb{R}^n$ n is the input vector, and $𝐴$ is the transformation matrix.

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Example 1:

If $T: \mathbb{R}^2 \to \mathbb{R}^2$ is defined as $T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x \\ 3y \end{bmatrix}$

it is a linear transformation because it satisfies both vector addition and scalar multiplication.

Example 2:

(Simple Scaling Transformation)

If $T: \mathbb{R}^2 \to \mathbb{R}^2$ scales a vector $\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}$, then: $T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x \\ 3y \end{bmatrix}$

For $\mathbf{x} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$, the output is: $T\left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 2 \cdot 1 \\ 3 \cdot 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 6 \end{bmatrix}$

Visualization of a Scaling Transformation

Scaling transforms a square grid, stretching it vertically and horizontally:

Original Grid	Scaled Grid (2x, 3y)

Step by step explanation

1. Original Grid:

   • A standard coordinate system with equal spacing

   • The point (1,2) marked in red

   • Grid lines for reference

2. Scaled Grid:

   • The same coordinate system after applying the transformation

   • The transformed point (2,6) marked in blue

   • Grid lines showing the scaling effect (2x horizontal, 3y vertical)

   • Reference axes remaining in original position

You can clearly see how the transformation stretches the grid, with:

• Horizontal spacing doubled (2x scaling in x-direction)

• Vertical spacing tripled (3y scaling in y-direction)

The example point moves from (1,2) to (2,6), demonstrating how the transformation affects individual points in the space.

Example 3:

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be defined as: $T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 3x + 2y \\ -x + 4y \end{bmatrix}.$

This transformation can be expressed using a matrix: $A = \begin{bmatrix} 3 & 2 \\ -1 & 4 \end{bmatrix}.$

Given , the transformation is:

$T(x) = Ax = \begin{bmatrix} 3 & 2 \\ -1 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

Perform the multiplication:

$T(x) = \begin{bmatrix} 3(1) + 2(2) \\ -1(1) + 4(2) \end{bmatrix} = \begin{bmatrix} 7 \\ 7 \end{bmatrix}.$

Visualization of the Transformation

Original Vector	Transformed Vector

The original grid is distorted based on the transformation matrix $A$, stretching and rotating the space.

Step by step explanation

Let's start with the given transformation matrix A and walk through how it transforms [1, 2] to [7, 7].

1. The transformation matrix A is:

   A = [3  2]
       [-1 4]

2. When we multiply matrix A by vector [1, 2], we get:

   [3  2] [1] = [3(1) + 2(2)]
   [-1 4] [2]   [-1(1) + 4(2)]

3. Let's calculate each component:

   • First component (x-coordinate):

     • 3(1) + 2(2)

     • = 3 + 4

     • = 7

   • Second component (y-coordinate):

     • -1(1) + 4(2)

     • = -1 + 8

     • = 7

4. Therefore:

   A[1] = [7]
    [2]   [7]

This shows how the linear transformation A maps the vector [1, 2] to [7, 7]. The transformation stretches and rotates the original vector in such a way that the resulting vector has coordinates [7, 7].

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2. Mathematical Properties of Linear Transformations

A linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ has the following properties:

1. Zero Vector Mapping: The zero vector in $V$ always maps to the zero vector in $W$: $T(0) = 0$\

2. Preservation of Linear Combinations: For vectors $u, v \in V$ and scalars $a, b \in \mathbb{R}$: $T(au + bv) = aT(u) + bT(v)$\

3. Kernel (Null Space): The set of all vectors that map to the zero vector: $\text{Ker}(T) = \{ x \in V : T(x) = 0 \}$\

4. Image (Range): The set of all vectors in $W$ that are outputs of $T$: $\text{Im}(T) = \{ T(x) : x \in V \}$

3. Geometric Interpretation

• Scaling stretches or compresses vectors.

• Rotation changes the direction of vectors.

• Reflection mirrors vectors across an axis.

Visualization

Below are visualizations of common transformations:

Scaling Transformation	Rotation Transformation	Reflection Transformation

4. Application of Matrices in Transforming Data

Linear transformations can be efficiently represented as matrix multiplications. For a transformation $T$ represented by matrix $A$:

$T(x) = Ax$

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Example 1:

The rotation matrix rotates vectors by an angle $\theta$:

$A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

For $\theta = 90^\circ$:

$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

Rotating $x = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$:

$Ax = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

Original Grid	Rotated Grid (90°)