Chapter 05: Linear Transformations

Prerequisites: Vector Spaces (Ch04)

Chapter Outline: Definition of Linear Transformations → Linearity Criteria → Matrix Representation (Choice of Bases) → Kernel and Range → Injectivity, Surjectivity, and Isomorphisms → Operations on Transformations (Sum, Scalar Multiple, Composition) → Inverse Transformations → Matrix Representation under Change of Basis (Similarity) → Geometric Operators (Rotation, Reflection, Projection, Shear)

Extension: Linear transformations connect static vector spaces and are the concrete manifestation of "morphisms" in the category of vector spaces; they are the core for understanding Diagonalization (Ch06) and SVD (Ch11).

Vector spaces provide the stage, and linear transformations are the primary actors. A linear transformation is a mapping that preserves the addition and scalar multiplication structures of a vector space, allowing us to study the interconnections between different spaces. This chapter reveals a central fact: once bases are chosen, every linear transformation can be uniquely represented by a matrix.

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05.1 Definition of Linear Transformations

Definition 05.1 (Linear Transformation)

A mapping \(T: V \to W\) is called a linear transformation if it satisfies: 1. Additivity: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\). 2. Homogeneity: \(T(c\mathbf{v}) = cT(\mathbf{v})\).

Properties

A linear transformation always maps the zero vector to the zero vector: \(T(\mathbf{0}_V) = \mathbf{0}_W\).

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05.2 Matrix Representation

Theorem 05.1 (Matrix Representation)

Let \(B = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\) be a basis for \(V\) and \(C\) be a basis for \(W\). \(T\) is completely determined by its action on the basis vectors \(T(\mathbf{v}_j)\). The matrix representation of \(T\) is \([T]_{C \leftarrow B} = [ [T(\mathbf{v}_1)]_C \ \cdots \ [T(\mathbf{v}_n)]_C ]\). Then: \([T(\mathbf{x})]_C = [T]_{C \leftarrow B} [\mathbf{x}]_B\).

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05.3 Kernel and Range

Definition 05.2 (Kernel and Range)

1. Kernel \(\ker(T)\): The set of all vectors mapped to the zero vector: \(\{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}\).
2. Range \(\operatorname{im}(T)\): The set of all images of vectors in \(V\): \(\{T(\mathbf{v}) : \mathbf{v} \in V\}\).

Theorem 05.2 (Rank-Nullity Theorem for Transformations)

\(\dim \ker(T) + \dim \operatorname{im}(T) = \dim V\). This is essentially identical to the rank-nullity theorem for matrices.

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05.4 Isomorphisms and Inverses

Definition 05.3 (Isomorphism)

If \(T: V \to W\) is both injective (one-to-one) and surjective (onto), it is an isomorphism. In this case, \(V\) and \(W\) are algebraically equivalent. Corollary: Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.

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Chapter Summary

Linear transformations build "bridges" between spaces:

****: The essence of a linear transformation is maintaining the harmony of vector addition and scalar multiplication.

****: Choosing a basis translates abstract mappings into matrix arithmetic, enabling the use of numerical algorithms for abstract logic.

****: The relationship between kernel and range (Rank-Nullity Theorem) reveals the patterns of information retention and loss during transformation.