Linear Transformation

Linear Transformations

A Linear Transformation $T: V \to W$ is a function between vector spaces that preserves the structure of the space.

Defining Properties

A map $T$ is linear if:

$T(u + v) = T(u) + T(v)$
$T(cu) = cT(u)$

Types of Maps

Injective (One-to-One): Kernel is only $\{0\}$ .
Surjective (Onto): Image is all of $W$ .
Isomorphism: Both Injective and Surjective.

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Kernel and Image

Kernel (Null Space): Everything that gets smashed to zero. $\text{Ker}(T) = \{ v \mid T(v) = 0 \}$ .
Image (Range): The actual output space. $\text{Im}(T) = \{ T(v) \mid v \in V \}$ .

Matrix Representation

Every linear transformation between finite-dimensional spaces can be represented as a matrix .

Basis of V

Transformation Matrix

Basis of W

All Chapters in this Book

Lesson 1

Vector and Matrices

Introduction to vectors, matrices, and their fundamental operations in linear algebra.

Lesson 2

Solving Systems of Linear Equations

Mastering techniques to solve linear systems: Cramer's Rule, Inverse Matrix, and Gauss Elimination.

Lesson 3

Introduction to Vector Space

Formal definition of vector spaces, axioms, and subspaces.

Lesson 4

Basis and Dimension

Understanding the building blocks of vector spaces: Linear Independence, Spanning Sets, Basis, and Dimension.

Lesson 5

Rank and Nullity

Exploring the fundamental subspaces of a matrix and the Rank-Nullity Theorem.

Lesson 6