Linear Transformations

In the study of vector spaces, the objects of interest are not merely the spaces themselves, but the mappings between them that preserve their algebraic structure. These mappings, known as linear transformations (or linear maps), form the foundation of functional analysis and numerical linear algebra.

1. Formal Definition

Let $V$ and $W$ be vector spaces over the same field $\mathbb{F}$. A function $T:V\to W$ is a linear transformation if, for all $u,v\in V$ and all $c\in \mathbb{F}$, the following two conditions hold:

1. Additivity: $T(u+v)=T(u)+T(v)$
2. Homogeneity: $T(cv)=cT(v)$

These conditions are equivalent to the single requirement of linearity: $T(cu+v)=cT(u)+T(v)$ for all $u,v\in V$ and $c\in \mathbb{F}$. We denote the set of all such maps as $\mathcal{L}(V,W)$. When $V=W$, we call $T$ a linear operator.

2. Kernel and Image

Associated with every linear map $T\in \mathcal{L}(V,W)$ are two fundamental subspaces:

The Kernel

The kernel (or null space) of $T$, denoted by $\ker (T)$, is the set of all vectors in $V$ that map to the zero vector in $W$: $\ker (T)=\{v\in V:T(v)=0_W\}$ The kernel measures the “loss of information” under $T$. $T$ is injective if and only if $\ker (T)=\{0_V\}$.

The Image

The image (or range) of $T$, denoted by $\text{im}(T)$ or $T(V)$, is the set of all vectors in $W$ that are reached by applying $T$ to vectors in $V$: $\text{im}(T)=\{T(v):v\in V\}$ $T$ is surjective if $\text{im}(T)=W$.

The First Isomorphism Theorem

A profound result in abstract algebra specialized for vector spaces states that: $V/\ker (T)\cong \text{im}(T)$ This implies the Rank-Nullity Theorem: If $V$ is finite-dimensional, then: $\dim (V)=\dim (\ker T)+\dim (\text{im}T)$ where $\dim (\text{im}T)$ is often called the rank of $T$ and $\dim (\ker T)$ the nullity.

3. The Matrix of a Transformation

In finite-dimensional spaces, every linear transformation can be represented as a matrix. Let $\mathcal{B}=\{v_1,\dots ,v_n\}$ be a basis for $V$ and $\mathcal{C}=\{w_1,\dots ,w_m\}$ be a basis for $W$. The matrix of $T$ with respect to bases $\mathcal{B}$ and $\mathcal{C}$, denoted by $[T{]}_{\mathcal{B}}^{\mathcal{C}}$, is the $m\times n$ matrix whose $j$-th column is the coordinate vector of $T(v_j)$ with respect to $\mathcal{C}$: $[T{]}_{\mathcal{B}}^{\mathcal{C}}=\left(\begin{array}{ccc}|& & |\\ [T(v_1){]}_{\mathcal{C}}& \dots & [T(v_n){]}_{\mathcal{C}}\\ |& & |\end{array}\right)$ For any $v\in V$, the transformation is equivalent to matrix-vector multiplication: $[T(v){]}_{\mathcal{C}}=[T{]}_{\mathcal{B}}^{\mathcal{C}}[v{]}_{\mathcal{B}}$

4. Isomorphisms and Coordinates

A linear map $T:V\to W$ is an isomorphism if it is bijective (both injective and surjective). If such a map exists, $V$ and $W$ are said to be isomorphic, written $V\cong W$.

Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension. Specifically, any $n$-dimensional vector space $V$ over $\mathbb{F}$ is isomorphic to ${\mathbb{F}}^n$ via the coordinate map ${\varphi }_{\mathcal{B}}:v\mapsto [v{]}_{\mathcal{B}}$.

5. Composition and Invertibility

Linear transformations can be composed. If $T:U\to V$ and $S:V\to W$ are linear maps, then $S\circ T:U\to W$ is also linear.

In terms of matrices, composition corresponds to matrix multiplication. If $\mathcal{A},\mathcal{B},\mathcal{C}$ are bases for $U,V,W$ respectively: $[S\circ T{]}_{\mathcal{A}}^{\mathcal{C}}=[S{]}_{\mathcal{B}}^{\mathcal{C}}[T{]}_{\mathcal{A}}^{\mathcal{B}}$ A map $T\in \mathcal{L}(V,W)$ is invertible if and only if its matrix $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ is invertible (square and non-singular).

6. Special Transformations

Scaling (Dilation)

A scaling transformation $S_{\mathbf{c}}\in \mathcal{L}({\mathbb{R}}^n)$ stretches or shrinks space along the principal axes. Given a vector of scales $\mathbf{c}=(c_1,c_2,\dots ,c_n)$, the transformation is represented by a diagonal matrix: $[S_{\mathbf{c}}]=\left(\begin{array}{cccc}{c}_1& 0& \dots & 0\\ 0& c_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & c_n\end{array}\right)$ If $c_i=c$ for all $i$, the map is a homothety or uniform scaling.

Projections (Idempotent Operators)

An operator $P\in \mathcal{L}(V)$ is a projection if $P^2=P$. Any such operator induces a direct sum decomposition: $V=\ker (P)\oplus \text{im}(P)$ Every vector $v\in V$ can be uniquely written as $v=(v-Pv)+Pv$, where $(v-Pv)\in \ker (P)$ and $Pv\in \text{im}(P)$.

Rotations

In ${\mathbb{R}}^2$, a rotation by angle $\theta$ counter-clockwise is given by: $R_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ Rotations are examples of orthogonal transformations, which preserve the inner product: $⟨Tv,Tw⟩=⟨v,w⟩$. In ${\mathbb{R}}^n$, they belong to the special orthogonal group $SO(n)$.

7. Invariant Subspaces

A subspace $U\subseteq V$ is invariant under $T\in \mathcal{L}(V)$ if $T(u)\in U$ for all $u\in U$. The study of invariant subspaces is central to canonical forms (like Jordan Normal Form). If $U$ is invariant, we can consider the restricted map $T{|}_U:U\to U$, which simplifies the analysis of $T$. Eigenvectors span the simplest invariant subspaces: one-dimensional lines.

8. The Adjoint of a Map (Briefly)

In an inner product space, for every linear map $T:V\to W$, there exists a unique map $T^{\ast }:W\to V$ called the adjoint such that: $⟨Tv,w⟩=⟨v,T^{\ast }w⟩$ for all $v\in V,w\in W$. In the standard basis of ${\mathbb{C}}^n$, the matrix of $T^{\ast }$ is the conjugate transpose of the matrix of $T$.

Python Implementation: Symbolic Linear Algebra

Using sympy, we can analyze linear transformations symbolically without numerical precision issues.

import sympy as sp

# Define symbols and a vector space basis
x, y, z = sp.symbols('x y z')
v = sp.Matrix([x, y, z])

# Define a linear transformation T: R^3 -> R^2
# T(x, y, z) = (x + 2y, y - z)
T_matrix = sp.Matrix([
    [1, 2, 0],
    [0, 1, -1]
])

# 1. Apply transformation to a general vector
result = T_matrix * v
print(f"T(x,y,z) = {result}")

# 2. Find the Kernel (Nullspace)
kernel = T_matrix.nullspace()
print(f"Basis for Kernel: {kernel}")

# 3. Find the Image (Column Space)
image = T_matrix.columnspace()
print(f"Basis for Image: {image}")

# 4. Verify Rank-Nullity
rank = T_matrix.rank()
nullity = len(kernel)
dim_V = T_matrix.cols
print(f"Rank ({rank}) + Nullity ({nullity}) = {dim_V}")

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