Introduction to Linear Transformations
The maps that respect addition and scaling — and the surprising fact that every single one of them is secretly just a matrix in disguise
§1Quick Recall — and a New Direction
One loose thread before we move on. Diagonalization can fail when an eigenvalue repeats — say $\lambda=2$ is a root of the characteristic polynomial twice (its algebraic multiplicity is $2$). Whether the matrix is still diagonalizable depends on whether the corresponding eigenspace is also $2$-dimensional (its geometric multiplicity) — if the eigenspace comes up "too small," the matrix is not diagonalizable at all. This is a real subtlety worth flagging, though today's lecture goes elsewhere; file it away for when repeated eigenvalues actually show up in a problem.
§2Definition of a Linear Transformation
A function $L : V \to W$ between two vector spaces is a linear transformation if it satisfies both:
$$L(v_1+v_2) = L(v_1)+L(v_2), \qquad \forall\, v_1,v_2 \in V \quad \textbf{(additivity)}$$
$$L(\alpha v) = \alpha L(v), \qquad \forall\, \alpha\in\mathbb{R},\; v\in V \quad \textbf{(homogeneity)}$$
In words: a linear transformation doesn't care whether you combine vectors first and then map them, or map them first and then combine — the two operations commute. This is the exact same pair of rules you've already used dozens of times when checking that a set is a subspace; now we're checking that a function respects that structure.
§3Three Basic Examples
- 1$L = \operatorname{id}$, the identity map: $L(v)=v$ for every $v$. Trivially linear.
- 2$L = 0$, the zero map: $L(v)=\mathbf{0}$ for every $v$. Also trivially linear.
- 3$L : V\to V$ given by $L(v) = c\cdot v$ for some fixed scalar $c$ — uniform scaling. Linear for every choice of $c$ (check: $c(v_1+v_2)=cv_1+cv_2$ and $c(\alpha v)=\alpha(cv)$, both immediate from ordinary scalar-vector algebra).
§4Every Linear Transformation Sends 0 to 0
$$L(\mathbf{0}) = \mathbf{0} \quad \text{always.}$$ One line proves it: $L(\mathbf{0}) = L(0\cdot v) = 0\cdot L(v) = \mathbf{0}$, using homogeneity with $\alpha=0$.
§5Composing Linear Transformations
Given $L : V\to W$ and $T : W \to X$, their composition is the map
$$T\circ L(v) := T(L(v)), \qquad v \in V.$$
Feed a vector into $L$, land in $W$, then feed that result into $T$ to land in $X$. The composition of two linear transformations is itself linear (a short, mechanical check using the same additivity/homogeneity rules on each map in turn) — a fact that will matter as soon as you start chaining transformations together, exactly the way you already chain matrix multiplications.
§6The Big Structural Question
Here is the question this whole lecture is really building toward: can we comprehend all possible linear transformations operating between a space $V$ of dimension $n$ and a space $W$ of dimension $m$?
$$T : V_n \longrightarrow W_m$$
$$T(v) = \underset{m\times n}{A}\;\underset{n\times1}{v} \;\in\; W, \qquad v\in V$$
— once you fix a basis for $V$ and a basis for $W$ (which basis, and how the matrix depends on that choice, is exactly the subject of Lecture 21). For today: the existence of that matrix is the point.§7Every Matrix Gives Us a Linear Transformation
Every matrix gives us a linear transformation.
This is the flip side of the Big Question above, and together the two directions say something remarkable: matrices and linear transformations are the same object, just viewed from two angles. Any $m\times n$ matrix $A$ defines a linear transformation $T(v)=Av$ from $\mathbb{R}^n\to\mathbb{R}^m$ (linearity of matrix multiplication — $A(v_1+v_2)=Av_1+Av_2$ and $A(\alpha v)=\alpha Av$ — is something you've been using since Lecture 5, you just hadn't named it "linear transformation" yet). Conversely, every linear transformation between finite-dimensional spaces comes from some matrix. Neither direction is the "real" one — they're the same fact, told twice.
§8The Derivative as a Linear Transformation
Differentiation, applied to functions rather than equations, has exactly the property we need:
$$\frac{d}{dx}\big(a\,f(x)+b\,g(x)\big) = a\,\frac{d}{dx}f(x) + b\,\frac{d}{dx}g(x).$$
$\blacksquare$ Differentiation is a linear transformation. (By the same reasoning — the integral of a sum is the sum of the integrals, and constants pull straight out — integration is linear too.)
§9A Formal Example — The Derivative Operator on Polynomials
Define $D:P_3\to P_2$ by $$D(a_0+a_1x+a_2x^2+a_3x^3) = a_1 + 2a_2x + 3a_3x^2.$$ Since $D$ is linear, $D$ is a linear transformation — no further check required beyond confirming the two defining rules, which is exactly the sum/constant rule from §8.
Relative to the standard bases $\{1,x,x^2,x^3\}$ of $P_3$ and $\{1,x,x^2\}$ of $P_2$, coordinates $(a_0,a_1,a_2,a_3)$ map to $(a_1,2a_2,3a_3)$, which is exactly matrix–vector multiplication by
$$D = \begin{pmatrix} 0&1&0&0\\ 0&0&2&0\\ 0&0&0&3 \end{pmatrix}.$$
Check the shape: $P_3$ has dimension $4$, $P_2$ has dimension $3$, so $D$ is a $3\times4$ matrix — exactly matching the $m\times n$ pattern from §6 ($n=4$, $m=3$). And the arithmetic checks out directly: $D\cdot(a_0,a_1,a_2,a_3)^T = (a_1,\,2a_2,\,3a_3)^T$, which is precisely $a_1+2a_2x+3a_3x^2$ written in coordinates.
§10A Geometric Example — Reflection Across the x-Axis
Define $L:\mathbb{R}^2\to\mathbb{R}^2$ by $L(x,y)=(x,-y)$ — flip every point to its mirror image across the $x$-axis. In matrix form:
$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}x\\-y\end{pmatrix}, \qquad L = \begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$
A geometric operation — reflecting — turns out to be nothing more than multiplication by a fixed $2\times2$ matrix.
Reflection across the x-axis keeps the x-coordinate and flips the sign of the y-coordinate — additive and scalar structure both survive the flip, which is exactly what makes it linear.
§11Practice Set — Is It a Linear Transformation?
For each map below, check additivity and homogeneity, then classify it as linear or not. Try each one before revealing the discussion.
$T:\mathbb{R}^3\to\mathbb{R}$, with $w=(1,0,1)$ fixed: $$T(v) = v\cdot w.$$
$T:M_{2\times2}\to\mathbb{R}$: $$T(A) = \det(A).$$
$T:\mathbb{R}\to\mathbb{R}$: $$T(x) = x^2.$$
$T:M_{2\times2}\to\mathbb{R}$: $$T(A) = \operatorname{tr}(A).$$
Cf. Nicholson §2.6 — Linear Transformations in ℝⁿ.
Additivity and homogeneity are the only two rules that matter — and every function obeying them, however exotic, turns out to be a matrix wearing a disguise.
Differentiation, reflection, projection, rotation, dot products with a fixed vector — wildly different-looking operations, all secretly the same kind of object. That unification is the entire point of calling them all "linear transformations" instead of studying each one separately.
For the upcoming Quiz and Exam: sections §7.1, §7.2.
There is a double quiz on Monday, covering §8.1, §8.2, §7.1, and §7.2.
A policy referred to in passing as the "n−2 policy" was also mentioned — see the official course policy document for the exact details rather than relying on this summary.