Language

This page collects the general vocabulary that the chapters assume before their own terminology begins. These words are easy to skim past, but confusing two of them can make a straightforward question feel opaque. Each chapter defines its own terms where they first appear.

Objects, sets and membership

• Mathematical object: anything being studied, such as a number, vector, matrix, polynomial, function, sequence or set. The word deliberately avoids committing to a particular kind.
• Set: a collection of distinct objects. Order and repetition do not matter. The notation $x\in S$ means “$x$ is an element of $S$”.
• Element or member: one object belonging to a set. $x\in S$ and $S\ni x$ mean the same thing.
• Subset: $A\subseteq B$ means every element of $A$ is also an element of $B$; equality is allowed. Some books write $A\subset B$ for a proper subset, so check the local convention.
• Proper subset: a subset that is not the whole set, written $A\subsetneq B$.
• Empty set: the set $\varnothing$ with no elements. It is a subset of every set.
• Set-builder notation: $\{x\in X: P(x)\}$ is the set of elements of $X$ for which the condition $P(x)$ holds. A vertical bar may replace the colon.
• Union: $A\cup B$ contains the elements in $A$, in $B$, or in both.
• Intersection: $A\cap B$ contains the elements common to both sets.
• Cartesian product: $A\times B$ is the set of ordered pairs $(a,b)$ with $a\in A$ and $b\in B$. The plane $\mathbb R^2$ is $\mathbb R\times\mathbb R$.
• Family: an indexed collection, such as $\{v_1,\ldots,v_k\}$ or $\{f_t:t\ge0\}$. Unlike a plain set, the indexing can make order or repetition meaningful.

Numbers and number systems

• Natural numbers $\mathbb N$: in these notes, $\mathbb N=\{0,1,2,\ldots\}$. This suits recurrences that start at $n=0$ and Taylor sums that begin with their constant term. A sum or sequence may still start at $n=1$ or another index; that starting index is part of the displayed definition, not a different convention for $\mathbb N$.
• Integers $\mathbb Z$: the positive and negative whole numbers together with zero.
• Rational numbers $\mathbb Q$: ratios $p/q$ of integers with $q\ne0$.
• Real numbers $\mathbb R$: the numbers on the continuous number line, both rational and irrational. They are the usual scalars for calculus.
• Complex numbers $\mathbb C$: numbers $a+bi$ with $a,b\in\mathbb R$ and $i^2=-1$. They include the real numbers and every nonconstant complex polynomial factors completely over them.
• Scalar: a number used to scale another object, usually real or complex in this course. “Scalar” describes a role, not a new number system.
• Coefficient: a scalar multiplying a term, vector or basis element.
• Constant: a quantity held fixed within the current problem. It may still be an unspecified parameter.
• Parameter: a symbol that labels a family of problems or objects. It is usually held fixed while the other variables are analysed and one then asks how the result changes as the parameter changes.

The number systems are nested:

Variables, indices and dimensions

• Variable: a symbol that can take values in a specified domain.
• Independent variable: the input of a function, often $x$ or time $t$.
• Dependent variable: a quantity determined by the independent variable, such as $y(t)$.
• Dummy variable: a bound symbol whose name carries no meaning, as in $\int_0^1 f(x)\,dx=\int_0^1 f(t)\,dt$.
• Index: a label such as $n$, $i$ or $j$ that selects an entry or term. It need not represent a physical quantity.
• Dimension: the meaning depends on context. The dimension of a vector space is the number of vectors in a basis; the dimensions of a matrix are its row and column counts. A domain $D\subseteq\mathbb R^n$ has ambient dimension $n$; other notions, such as the dimension of a curve inside $\mathbb R^n$, must be specified separately.

Functions and maps

• Function or map: a rule that assigns exactly one output to each allowed input. The notation $f:X\to Y$ names the domain $X$ and codomain $Y$.
• Domain: the set of allowed inputs.
• Codomain: the declared set in which outputs live. It is part of the function's specification.
• Image or range: the outputs actually produced, $\operatorname{im}f=\{f(x):x\in X\}$. It is a subset of the codomain.
• Image of a subset: for $A\subseteq X$, $f(A)=\{f(a):a\in A\}$.
• Preimage or inverse image: for $B\subseteq Y$, $f^{-1}(B)=\{x\in X:f(x)\in B\}$. This makes sense even when $f$ has no inverse function.
• Formula: an expression that describes a function or quantity. A function is more than its formula: the same formula on a different domain or codomain can define a different function.
• Composition: $(g\circ f)(x)=g(f(x))$. The outputs of $f$ must be valid inputs for $g$. Composition is usually not commutative.
• Identity map: $\operatorname{id}_X(x)=x$. Composing a map with it changes nothing.
• Restriction: the same rule applied on a smaller domain.
• Injective or one-to-one: different inputs give different outputs.
• Surjective or onto: every element of the codomain is attained.
• Bijective: both injective and surjective. These are exactly the functions that have an inverse function on their stated domain and codomain.

Expressions, equations and identities

• Expression: a mathematical phrase such as $x^2+1$. It has a value once its variables are specified, but makes no claim by itself.
• Equation: a claim that two expressions are equal under stated conditions. Solving it means finding the values that make the claim true.
• Identity: an equality that holds for every value in its stated domain, such as $\sin^2x+\cos^2x=1$. An equation may hold only for selected values.
• Inequality: a comparison using $<,\le,>,\ge$. Bounds and estimates are written as inequalities.
• System: several equations or conditions required at once.
• Solution: an object that satisfies every condition in the problem.
• Solution set: the collection of all solutions. Finding one solution does not solve the problem unless uniqueness is known.
• Constraint: a condition that restricts the allowed candidates.
• Homogeneous equation: in a linear setting, an equation with zero on the forced side, such as $Ax=0$ or $L[y]=0$.
• Inhomogeneous, nonhomogeneous or forced equation: a linear equation with a nonzero right-hand side, such as $Ax=b$ or $L[y]=g$.

Logic words inside ordinary sentences

• For all $\forall$: every allowed object has the property. One counterexample defeats the statement.
• There exists $\exists$: at least one object has the property. This does not assert uniqueness.
• There exists a unique $\exists!$: exactly one object has the property.
• Implies $\Rightarrow$: whenever the first statement is true, so is the second.
• If and only if $\Leftrightarrow$: each statement implies the other. Often abbreviated “iff”, it expresses an equivalence.
• Necessary condition: $Q$ is necessary for $P$ when $P\Rightarrow Q$. Without $Q$, $P$ cannot occur.
• Sufficient condition: $P$ is sufficient for $Q$ when $P\Rightarrow Q$. $P$ guarantees $Q$, though it may not be the only way to obtain $Q$.
• Equivalent conditions: conditions that imply one another and so describe the same class of objects from different viewpoints.
• Contrapositive: $P\Rightarrow Q$ is logically equivalent to $\neg Q\Rightarrow\neg P$. It is not the converse $Q\Rightarrow P$, which may be false.
• Without loss of generality: a symmetry or relabelling makes the omitted cases genuinely equivalent to the one considered. It is not permission to ignore inconvenient cases.

Definitions, theorems and proofs

• Definition: a convention that fixes the meaning of a term. A definition is not proved, though one may prove that an object satisfies it.
• Axiom: a foundational rule accepted within a mathematical system. The vector-space axioms, for example, specify how the two operations must behave.
• Theorem: a claim proved from definitions, axioms and earlier results.
• Proposition: usually a theorem of more local scope, with the same mathematical status.
• Lemma: a result proved mainly to help prove another.
• Corollary: a result that follows quickly from a previous theorem.
• Hypothesis or assumption: a condition taken as true in a theorem or argument.
• Conclusion: what follows when the hypotheses hold.
• Proof: a logical argument showing that the conclusion follows from the hypotheses.
• Example: an object that illustrates a possibility. It does not prove a universal claim.
• Counterexample: one example that satisfies the hypotheses of a proposed claim but violates its conclusion. It disproves a universal claim.
• Conjecture: a claim believed to be true but not yet proved in the current context.

Operations and structural adjectives

• Unary operation: takes one input, such as complex conjugation.
• Binary operation: combines two inputs, such as addition or matrix multiplication.
• Closed under an operation: applying the operation to allowed elements produces another allowed element.
• Commutative: order does not matter: $a*b=b*a$.
• Associative: grouping does not matter: $(a*b)*c=a*(b*c)$. This does not permit changing the order.
• Distributive: one operation spreads over another, as multiplication does over addition.
• Linear: preserves addition and scalar multiplication. Informally, it has no interaction terms, powers or offsets relative to the stated vector-space structures.
• Affine: linear plus a fixed translation. The map $x\mapsto Ax+b$ is affine and linear only when $b=0$.
• Nonlinear: not linear. This does not mean random or unsolvable; it means superposition cannot be assumed.
• Invariant: kept within itself by an operation or map. An invariant subspace may contain vectors that move, provided they stay in the subspace.
• Symmetric: unchanged by a stated interchange or transformation. The word has several precise uses; a symmetric matrix satisfies $A^{\mathsf T}=A$.

Exact, approximate and numerical

• Exact value: a representation equal to the quantity itself, such as $\sqrt2$ or $\pi/4$, rather than a rounded decimal.
• Approximation: a simpler quantity used in place of the exact one, within an understood context or error.
• Error: exact value minus approximate value, with a sign convention that should be stated. Absolute error is the magnitude of this difference; relative error divides that magnitude by a nonzero exact or reference magnitude.
• Error bound: a guaranteed upper bound on the magnitude of an error. It need not equal the actual error.
• Numerical method: an algorithm that produces approximations through arithmetic, often using a step size or a finite truncation.
• Symbolic method: manipulates formulas to obtain an exact expression when possible.
• Truncation: stopping an infinite process after finitely many terms or steps.
• Rounding: replacing numbers by finite-precision representations.
• Order notation: $f(x)=O(g(x))$ near a limit point $a$ means there are constants $C,\delta>0$ such that $|f(x)|\le C|g(x)|$ whenever $0<|x-a|<\delta$ on the stated domain. It describes scale, not literal equality of formulas.

Local, global, discrete and continuous

• Local: concerning behaviour near one point or over a short time. A local maximum need not be the largest value on the whole domain.
• Global or absolute: concerning the entire stated domain.
• Discrete: indexed by separated steps, usually integers. Recurrences and sequences describe discrete evolution.
• Continuous: varying over an interval of real values. Differential equations describe continuous-time evolution.
• Finite: containing or using finitely many elements, terms or dimensions.
• Infinite: not finite. Infinite sums and limiting processes are defined through limits; they are not just very large finite calculations.
• Bounded: confined within finite limits. The precise meaning depends on whether the object is a set, function or sequence.
• Asymptotic behaviour: behaviour as a variable approaches a limiting regime, commonly $n\to\infty$, $t\to\infty$ or $x\to a$.

Distinctions that cut across chapters

Most easily confused pairs are separated in the lists above. Three more come up in every chapter:

When a symbol has several meanings

The same letter can name different kinds of object in different chapters:

• $i$: the imaginary unit or an integer index.
• $r$: a polar radius, a characteristic root or a geometric-series ratio.
• $\lambda$: an eigenvalue, a Lagrange multiplier or another parameter.
• $A$: a matrix, a set, a coefficient or an arbitrary constant.
• $D$: a region, a differentiation operator or a diagonal matrix.
• $R$: a region, a Taylor remainder or a radius of convergence.
• $P$: a polynomial, a probability, a population or a basis-change matrix.
• $|\cdot|$: real absolute value or complex modulus. These notes usually write determinants with $\det$ to keep the roles clear.

Before a long calculation, note each symbol's object type and its role in that problem. In particular, distinguish the polynomial space $P_n(F)$ from a Taylor polynomial $P_n(x)$ and a domain $D$ from its coordinate bounds.

Reading a new definition

Being able to repeat a definition is only the first layer. For each term, aim to know five things:

1. Definition: the precise condition that makes the term apply.
2. Picture: the informal idea the definition captures.
3. Test: how to decide whether an example has the property.
4. Connections: which earlier ideas it depends on and which later ideas use it.
5. Non-example: the nearest tempting idea that is not the same thing.

When a dense definition appears, rewrite it in five lines:

Object:              what kind of thing is being classified
Property:            the name of the property
Condition:           what the object must satisfy
Closest non-example: the nearest thing that fails the condition
Consequence:         what the property lets you do next

For example:

Object:              a list of vectors
Property:            linear independence
Condition:           only the zero coefficients give the zero linear combination
Closest non-example: a spanning list with a redundant vector
Consequence:         coefficients in that list are unique

The consequence is often the real reason the term was introduced.

Before calculating

1. Identify each object and its domain, codomain or feasible set.
2. Separate the hypotheses from the conclusion you need.
3. Decide whether the task asks for an exact value, an approximation or a numerical estimate.
4. Check whether a claim is local or global and whether the setting is discrete or continuous.
5. When changing coordinates, update the bases, bounds or area factor as the relevant chapter requires.
6. Check whether division, substitution or square roots have restricted the domain or removed possible solutions.
7. Decide whether a formula also needs an existence, uniqueness, convergence or error argument.