Mathematics Lecture Notes

Definition2.1Linear transformation

Let $V$ and $W$ be vector spaces over the same field $F$ . A function $T : V \to W$ is a linear transformation (or linear map) if for all $u, v \in V$ and $c \in F$ :

$T(u + v) = T(u) + T(v)$ (preserves addition)
$T(cv) = cT(v)$ (preserves scalar multiplication)

Equivalently, $T(au + bv) = aT(u) + bT(v)$ for all $a, b \in F$ and $u, v \in V$ .

RemarkNotation

We write $T : V \to W$ or $V \xrightarrow{T} W$ . The set of all linear transformations from $V$ to $W$ is denoted $\mathcal{L}(V, W)$ or $\text{Hom}_F(V, W)$ . When $V = W$ , we write $\mathcal{L}(V)$ and call $T$ a linear operator (or endomorphism).

RemarkT(0) = 0

Every linear map sends $0$ to $0$ : $T(0) = T(0 \cdot v) = 0 \cdot T(v) = 0$ . If a function fails this, it is not linear.

ExampleIdentity and zero maps

For any vector space $V$ :

The identity map $\text{id}_V : V \to V$ , $\text{id}_V(v) = v$ , is linear.
The zero map $T_0 : V \to W$ , $T_0(v) = 0$ , is linear.

ExampleMatrix multiplication

For $A \in M_{m \times n}(F)$ , the map $T_A : F^n \to F^m$ defined by $T_A(x) = Ax$ is linear. This is the most important example: every linear map between finite-dimensional spaces can be represented this way.

ExampleDifferentiation

$D : P_n(\mathbb{R}) \to P_{n-1}(\mathbb{R})$ defined by $D(p) = p'$ (the derivative) is linear:

$D(af + bg) = (af + bg)' = af' + bg' = aD(f) + bD(g).$

More generally, $D : C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ is a linear operator.

ExampleIntegration

$T : C[0,1] \to \mathbb{R}$ defined by $T(f) = \int_0^1 f(x)\,dx$ is linear:

$T(af + bg) = \int_0^1 (af + bg)\,dx = a\int_0^1 f\,dx + b\int_0^1 g\,dx = aT(f) + bT(g).$

ExampleProjection

$\pi : \mathbb{R}^3 \to \mathbb{R}^2$ defined by $\pi(x, y, z) = (x, y)$ (projection onto the $xy$ -plane) is linear. More generally, the projection onto any subspace along a complement is linear.

ExampleRotation in R^2

Rotation by angle $\theta$ about the origin is a linear map $R_\theta : \mathbb{R}^2 \to \mathbb{R}^2$ :

$R_\theta(x, y) = (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta).$

This corresponds to multiplication by the rotation matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ .

ExampleReflection

Reflection across the $x$ -axis in $\mathbb{R}^2$ : $T(x, y) = (x, -y)$ is linear. Reflection across any line through the origin is linear.

ExampleTranspose map

$T : M_{n \times n}(F) \to M_{n \times n}(F)$ defined by $T(A) = A^T$ is linear:

$(aA + bB)^T = aA^T + bB^T.$

ExampleTrace

$\text{tr} : M_{n \times n}(F) \to F$ defined by $\text{tr}(A) = \sum_{i=1}^n a_{ii}$ is linear. This is a linear functional (a linear map to the base field).

ExampleEvaluation map

For $c \in F$ , the map $\text{ev}_c : P(F) \to F$ defined by $\text{ev}_c(p) = p(c)$ is linear. This is another linear functional.

ExampleLeft shift

The left shift $L : F^\infty \to F^\infty$ defined by $L(a_1, a_2, a_3, \ldots) = (a_2, a_3, a_4, \ldots)$ is linear. It "deletes" the first entry. Similarly, the right shift $R(a_1, a_2, \ldots) = (0, a_1, a_2, \ldots)$ is linear.

ExampleNon-example: translation

$T : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $T(x, y) = (x + 1, y)$ is not linear because $T(0, 0) = (1, 0) \neq (0, 0)$ .

ExampleNon-example: squaring

$T : \mathbb{R} \to \mathbb{R}$ defined by $T(x) = x^2$ is not linear: $T(2 + 3) = 25 \neq 4 + 9 = T(2) + T(3)$ .

Theorem2.1Determined by values on a basis

Let $\beta = \{v_1, \ldots, v_n\}$ be a basis for $V$ , and let $w_1, \ldots, w_n$ be any vectors in $W$ (not necessarily distinct or independent). Then there exists a unique linear transformation $T : V \to W$ with $T(v_i) = w_i$ for all $i$ .

Proof

Define $T(a_1 v_1 + \cdots + a_n v_n) = a_1 w_1 + \cdots + a_n w_n$ . This is well-defined (since the $a_i$ are unique by linear independence) and linear (by construction). Uniqueness follows because any linear $S$ with $S(v_i) = w_i$ satisfies $S(a_1 v_1 + \cdots + a_n v_n) = a_1 w_1 + \cdots + a_n w_n = T(\cdot)$ .

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RemarkLinear maps form a vector space

$\mathcal{L}(V, W)$ is itself a vector space over $F$ under pointwise operations:

$(T + S)(v) = T(v) + S(v), \quad (cT)(v) = cT(v).$

If $\dim V = n$ and $\dim W = m$ , then $\dim \mathcal{L}(V, W) = mn$ .

RemarkComposition is linear

If $T : V \to W$ and $S : W \to U$ are linear, then $S \circ T : V \to U$ is linear. Composition is associative: $(R \circ S) \circ T = R \circ (S \circ T)$ .

RemarkLooking ahead

The key invariants of a linear map are its kernel and image. Their relationship is captured by the Rank-Nullity Theorem. In finite dimensions, every linear map can be encoded as a matrix.

Math Notes

Linear Transformations

Definition

Examples

Basic properties