3.1 Definition and Properties of Linear Transformations

Linear transformations are functions between vector spaces that preserve addition and scalar multiplication. They're crucial in understanding how vectors behave under different operations, forming the backbone of linear algebra and its applications in various fields.

This section dives into the definition, properties, and examples of linear transformations. We'll learn how to identify and compose them, exploring their unique characteristics and how they relate to other concepts in linear algebra.

Linear Transformations: Definition and Properties

Definition and Key Concepts

• A linear transformation (also known as a linear map or linear operator) is a function $T$ from one vector space $V$ to another vector space $W$ that preserves the operations of vector addition and scalar multiplication
• For any vectors $u$ and $v$ in $V$ and any scalar $c$, a linear transformation $T$ satisfies two properties: $T(u + v) = T(u) + T(v)$ (additivity) and $T(cu) = cT(u)$ (homogeneity)
• The domain of a linear transformation is the vector space $V$, and the codomain is the vector space $W$
• The image of $T$, denoted $Im(T)$ or $T(V)$, is the subset of $W$ consisting of all vectors that are outputs of $T$
• The kernel (or null space) of a linear transformation $T$, denoted $Ker(T)$ or $Null(T)$, is the set of all vectors $v$ in $V$ such that $T(v) = 0$, where $0$ is the zero vector in $W$
• A linear transformation maps the zero vector of $V$ to the zero vector of $W$, i.e., $T(0) = 0$

Properties and Uniqueness

• Additivity: For any vectors $u$ and $v$ in the domain $V$, $T(u + v) = T(u) + T(v)$ preserves vector addition
• Homogeneity: For any vector $u$ in $V$ and any scalar $c$, $T(cu) = cT(u)$ preserves scalar multiplication
• Preservation of the zero vector: $T(0) = 0$, where $0$ on the left is the zero vector in $V$ and $0$ on the right is the zero vector in $W$
• Uniqueness: A linear transformation is uniquely determined by its action on a basis of the domain vector space $V$

Identifying Linear Transformations

Verifying Linearity

• To prove that a function $T$ is a linear transformation, verify that it satisfies the additivity and homogeneity properties for any vectors $u$ and $v$ in the domain $V$ and any scalar $c$
• To show that a function is not a linear transformation, find a counterexample where either the additivity or homogeneity property fails to hold
• Common non-linear functions include absolute value, quadratic ($(x, y) \mapsto (x^2, y^2)$), exponential ($(x, y) \mapsto (e^x, e^y)$), and logarithmic ($(x, y) \mapsto (\ln x, \ln y)$) functions
• When working with functions between vector spaces of matrices or polynomials, the additivity and homogeneity properties must hold for matrix addition and multiplication or polynomial addition and multiplication, respectively

Examples of Linear Transformations

• The identity transformation $I: V \to V$, defined by $I(v) = v$ for all $v$ in $V$, is a linear transformation
• The zero transformation $Z: V \to W$, defined by $Z(v) = 0$ for all $v$ in $V$, where $0$ is the zero vector in $W$, is a linear transformation
• The projection onto the $x$-axis $P_x: \mathbb{R}^2 \to \mathbb{R}$, defined by $P_x(x, y) = x$, is a linear transformation
• The rotation by an angle $\theta$ in $\mathbb{R}^2$, defined by $R_\theta(x, y) = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$, is a linear transformation

Composing Linear Transformations

Composition and Linearity

• The composition of two linear transformations is a new linear transformation
• If $T: V \to W$ and $S: W \to U$ are linear transformations, then their composition $S \circ T: V \to U$, defined by $(S \circ T)(v) = S(T(v))$, is also a linear transformation
• To prove that the composition of two linear transformations is linear, let $u$ and $v$ be vectors in $V$ and $c$ be a scalar
  1. Show that $(S \circ T)(u + v) = (S \circ T)(u) + (S \circ T)(v)$ by using the linearity of $S$ and $T$ separately
  2. Show that $(S \circ T)(cu) = c(S \circ T)(u)$ by using the linearity of $S$ and $T$ separately

Properties of Composition

• The composition of linear transformations is associative: $(T \circ S) \circ R = T \circ (S \circ R)$ for linear transformations $R$, $S$, and $T$
• For any linear transformation $T: V \to W$, $T \circ I = T$ and $I \circ T = T$, where $I$ is the identity transformation on the appropriate vector space ($V$ or $W$)
• The composition of a linear transformation with the zero transformation results in the zero transformation: $T \circ Z = Z$ and $Z \circ T = Z$, where $Z$ is the zero transformation on the appropriate vector space