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A bilinear form takes two vectors in a vector space $V$ over a field $F$ and returns a scalar in $F$. Its defining property is separate linearity: holding either input fixed gives a linear functional of the other. For scalars $a,b$ and vectors $u,v,w$,This is a scalar-valued bilinear map; the chosen field is part of the definition.^[ref]

Separate linearity does not mean that $B$ is a linear map on the vector space $V\times V$ with componentwise vector addition and scalar multiplication. Over the real numbers, $B(2v,2w)=4B(v,w)$, whereas a linear map on the pair would give $2B(v,w)$. The simplest example is $B(x,y)=xy$ on the one-dimensional vector space $\mathbb{R}$: fixing $y=3$ gives $x\mapsto 3x$, but doubling both $x$ and $y$ quadruples their product.

A bilinear form measures how two inputs interact. The dot product measures alignment; a symmetric bilinear form can describe a quadratic form; an alternating bilinear form can measure signed area. A nondegenerate bilinear form turns each vector into a unique linear functional and so connects geometry with the dual space. The same short definition therefore supports lengths, bilinear orthogonal complements, and symplectic vector spaces, depending on the additional properties imposed.

Over the complex numbers, a bilinear form is linear in both inputs without conjugating either scalar. A Hermitian form instead is a sesquilinear form and conjugates when its inputs are exchanged. For complex matrices, the adjoint $A^{\ast }$ used with Hermitian forms and matrix C-star algebras is the conjugate transpose, whereas a bilinear form uses the ordinary transpose $A^T$. For example, $B(z,w)=zw$ has $B(i,i)=-1$, while the Hermitian form $H(z,w)=z\overline{w}$ has $H(i,i)=1$.^[ref]

A map $B:V\times W\to Z$ between vector spaces is bilinear when it is linear in either argument with the other held fixed. A scalar-valued bilinear form is the case $V=W$ and $Z$ is the scalar field. Checking bilinearity is one of the algebraic steps in verifying a real inner product; being a symmetric bilinear form and having positive squared norm impose further conditions.

A bilinear map is a map $B:V\times W\to U$ between vector spaces that is linear in each argument when the other is fixed. The nonlinear temperature advection can be written as the diagonal of a bilinear map, $\mathcal{N}(\varphi ,\chi )=-\mathbf{u}[\varphi ]\cdot \nabla \chi$, when velocity depends linearly on temperature. Unlike a bilinear form, its output need not be a scalar.

A block matrix is block diagonal if all off-diagonal blocks vanish. It represents independent actions on a direct sum of vector spaces. Its determinant is the product of the diagonal-block determinants.

Replace each positive power $x_i^d$ by $x_i$ in a polynomial. This gives a multilinear polynomial of no larger polynomial degree with the same values on $\{0,1{\}}^n$. In the polynomial method in combinatorics, these functions lie in a vector space with a basis consisting of the square-free monomials of the required polynomial degrees.

For a nonzero finite-dimensional complex vector space $V$, a unital matrix algebra acting irreducibly on $V$ is the full endomorphism algebra. Its Jacobson radical annihilates $V$: otherwise the radical times $V$ is all of $V$, contradicting nilpotence. Faithfulness kills the radical, and the Artin–Wedderburn theorem then leaves one simple matrix factor acting on its standard column module. This is a matrix-algebra theorem, distinct from the orbit-counting Burnside lemma and the finite-group Burnside's theorem.

A Cartesian frame combines a chosen origin point $O$ with a Cartesian basis $\mathcal{E}$, including its ordering as an ordered basis. A point $P$ has coordinate vector $c$ when $\overrightarrow{OP}={\sum }_i{c}_i{e}_i$. The basis supplies measuring directions and unit lengths; the coordinate vector is the resulting tuple of numbers; the Cartesian frame additionally fixes where measurement starts. A vector already belongs to a vector space with a zero vector, so expressing it in a basis requires no additional choice of spatial origin.

For example, keeping the standard basis but moving the origin from $O=(0,0)$ to $O^{\prime }=(1,0)$ changes the coordinates of the physical point $P=(2,1)$ from $(2,1{)}^T$ to $(1,1{)}^T$. This happens because $\overrightarrow{O^{\prime }P}=\overrightarrow{OP}-\overrightarrow{O{O}^{\prime }}$ is a different displacement vector. By contrast, the two diagrams above keep $O$ fixed and change only the basis. In a fixed reference Cartesian frame, a new frame with origin column $o$ and basis-column matrix $Q$ describes points by $p=o+Qc$; the new coordinates are $c=Q^T(p-o)$ when $Q$ is an orthogonal matrix.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

The two affine opens $D(x),D(y)$ cover the punctured affine plane. Their intersection has ring $k[x^{\pm 1},y^{\pm 1}]$. The Čech cohomology in degree one is the quotient by $k[x^{\pm 1},y]+k[x,y^{\pm 1}]$. Its basis consists precisely of the Laurent monomials whose two exponents are negative, so this vector space is infinite-dimensional. Global functions, by contrast, form $k[x,y]$.

A change of basis rewrites a vector using a different ordered basis of the same vector space. The vector stays fixed; its coordinate vector changes because the measuring directions change. Both bases must describe every vector uniquely, so a change of basis is reversible.

Start with the Cartesian basis $e_1=(1,0)$, $e_2=(0,1)$, and choose the new basis vectors $u=(1,1)$ and $v=(1,-1)$. For the vector $z=(2,0)$,The second coordinate vector says to travel once along $u$ and once along $v$. Their vertical contributions cancel, leaving exactly the original horizontal arrow. The diagram shows two descriptions of that same arrow, with the same origin and endpoint.

Choosing a useful basis can reveal simple structure hidden by the old coordinate vectors. Here the new basis describes common and opposing contributions. For two equal measurements $(5,5)$, the new coordinate vector is $(5,0)$: only the common contribution is present. In larger examples, a suitable basis can concentrate important information into a few coefficients. Strang's lecture on change of basis develops this principle through applications: the reversible change of basis preserves information, while subsequently discarding small coefficients loses information.^[ref]

A linear subspace $I$ of a coalgebra satisfying the displayed conditions. These are exactly the conditions for the comultiplication and counit to descend to the quotient vector space $C/I$. An ideal that is also a coideal in a bialgebra gives a quotient bialgebra.

The commutator of two elements of an associative algebra over a field isIt measures how much the order of multiplication matters: $[A,B]=0$ exactly when $AB=BA$. It is a bilinear map, it is alternating since $AA-AA=0$, and associativity makes it satisfy the Jacobi identity. Hence every associative algebra becomes a Lie algebra on the same vector space when equipped with this product. This holds in every characteristic of a field.

For square matrices, this is the matrix commutator, and it is defined even for singular matrices. The group commutator of invertible matrices is instead $AB{A}^{-1}{B}^{-1}$: it equals the identity element when the matrices commute, whereas $AB-BA$ equals the zero matrix. The Lie bracket from local group commutators explains how differentiation connects the two constructions.

A complete flag in an $n$-dimensional vector space is a nested sequence of vector subspaces with $\dim V_i=i$. An action preserves such a flag exactly when its matrices are upper triangular in a basis adapted to the flag. The Lie theorem supplies an invariant complete flag for every complex finite-dimensional Lie algebra representation of a solvable Lie algebra.

For nonzero finite-dimensional complex vector spaces $U,V$, suppose a bilinear map never vanishes on a pair of nonzero vectors. Equivalently, each map obtained by fixing one nonzero argument is injective. If its associated map on $U\otimes V$ has image dimension $r$, projectivization gives a map to ${\mathbb{CP}}^{r-1}$ whose positive degree-two class pulls back to $x+y$. The nonzero top power $(x+y{)}^{\dim U+\dim V-2}$ in the product cohomology ring of complex projective space proves the bound. Multiplication of polynomials of bounded degree attains equality. Nonzero hypotheses matter: with a zero factor, slice conditions can be vacuous.

A vector space over the field of complex numbers. A complex-linear map preserves addition and multiplication by complex scalars.

A type-$(1,1)$ tensor identifies with an endomorphism of a finite-dimensional vector space. Composing two such maps produces another type-$(1,1)$ tensor, because $(X,\eta )\mapsto \eta (B(A(X)))$ is a multilinear map. In output-first components the result is $C^a{}_b=B^a{}_c{A}^c{}_b$; the contracted index implements composition, and a basis change conjugates the corresponding matrices.

In an $N$-dimensional real vector space, each member of a conic hull has a representation using at most $N$ generators. To prove it, take a finite positive-coefficient representation with more than $N$ terms. Its generators are linearly dependent, say ${\sum }_j{\mu }_j{s}_j=0$, with some ${\mu }_j>0$ after reversing the relation if necessary. Subtract $t{\mu }_j$ from each coefficient, taking $t={\min }_{{\mu }_j>0}{\lambda }_j/{\mu }_j$. All coefficients stay nonnegative and at least one vanishes. Iterate. The bound differs from the $N+1$ bound for a convex hull because the coefficient sum is unrestricted.

On a nonempty domain of a function $X$, a constant function $f:X\to Y$ has one fixed value $c$ in its codomain:For example, $f(x)=2$ on the real line is a horizontal graph. Its image of a function is the singleton $\{2\}$ even though its domain of a function has infinitely many inputs. A constant function from the real line to the real numbers has derivative zero and every nonzero real period of a function; there is no smallest positive period. It supplies a reference level from which variation can be measured.

A constant map between vector spaces is a linear map only when its value is the zero vector, since a linear map must preserve zero. A nonzero constant map is still an affine map. In a Fourier series, the constant term records the period average, while the other terms describe oscillation around that level.

The continuous dual space $X^{\ast }$ of a topological vector space is the vector space of its continuous linear functionals. Unlike the algebraic dual, it requires that a small change in an input produce a small change in its scalar measurement. For a normed vector space, this is equivalent to a finite boundfor all vectors $x$ and some constant $C\ge 0$. The least such constant is the operator norm of $\phi$.

This is the continuity-restricted subspace of the algebraic dual space; in infinite dimension the two can differ.

For the linear functional $\phi (x,y)=2x-y$ on ${\mathbb{R}}^2$ with its Euclidean norm, the Cauchy-Schwarz inequality givesEquality holds on the unit vector $(2,-1)/\sqrt{5}$, so its operator norm is exactly $\sqrt{5}$. The same estimate controls measurement error: $|\phi (v+h)-\phi (v)|\le \sqrt{5}\Vert h{\Vert }_2$. The linear functional diagram shows which changes are entirely invisible to this measurement; its kernel of a linear map consists of the changes with zero response. All linear functionals on a finite-dimensional real or complex normed vector space are continuous, but discontinuous functional detected by vanishing sequence tails shows why the restriction matters in infinite dimension.

A contraction mapping $T:X\to X$ on a nonempty complete metric space has exactly one fixed point $x_{\ast }$, and the sequence $x_{n+1}=T(x_n)$ converges to $x_{\ast }$ from every $x_0\in X$. The same constant $0\le L<1$ must satisfy $d(Tx,Ty)\le Ld(x,y)$ for all $x,y\in X$. The statement applies to metric spaces, including closed subsets of a Banach space; neither linearity nor a vector space structure is required.^[ref]

The force of the result is that existence, uniqueness, and a practical approximation procedure arrive together. Repeated application produces corrections bounded by a geometric series, hence a Cauchy sequence; completeness supplies its limit. The Lipschitz continuity of $T$ makes that limit a fixed point. Two different fixed points would have their positive distance multiplied by at most $L<1$ while remaining unchanged, which is impossible.