On a smooth variety over a field of characteristic zero, a general member of a basepoint-free linear system of divisors is smooth. More generally it is smooth outside the base locus. Irreducibility is a separate assertion and requires suitable hypotheses on the associated morphism; a pencil composed with a map to a curve can have disconnected general divisors.

A focus is a planar equilibrium whose linearization has nonreal complex-conjugate eigenvalues with nonzero real part. The imaginary part produces local rotation, while the real part determines growth or decay of its envelope. A stable focus has negative real part; a focus with positive real part repels nearby trajectories.

The linear system $\dot{x}=-\alpha x-\omega y$, $\dot{y}=\omega x-\alpha y$ with $\alpha >0$ and $\omega \ne 0$ has radius $r(t)=r(0)e^{-\alpha t}$ and angular advance $\omega t$. It therefore spirals inward, combining two measurable scales: rotation period $2\pi /|\omega |$ and decay time $1/\alpha$. A damped mechanical oscillator can exhibit this geometry in its position-velocity plane. Eigenvalues with zero real part require a separate nonlinear analysis and are excluded from this hyperbolic focus definition.

A node is a planar hyperbolic equilibrium whose two linearized eigenvalues are real and have the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Nearby motion approaches or recedes without the repeated rotation characteristic of a focus.

For $\dot{x}=-ax$, $\dot{y}=-by$ with $a,b>0$, solutions are $x=x_0{e}^{-at}$ and $y=y_0{e}^{-bt}$. Each coordinate decays, while unequal decay rates usually make a trajectory align with the slower direction as it nears the origin. Reversing both signs gives outward motion. Repeated eigenvalues require attention to the number of independent eigenvectors and can add polynomial factors to the exponential response. The node classification describes local behavior near an equilibrium; it does not determine the geometry of remote orbits.

In a normed vector space, the closed linear span of $S$ is the closure of its linear span in the norm topology. It includes limits of finite linear combinations. In a Hilbert space, an infinite orthonormal basis spans in this closed sense, allowing norm-convergent series in addition to finite sums.

In resistive-force theory with $0<\gamma <1$, head drag $\delta \ge 0$ and the ideal constant-inclination condition ${\alpha }^2=\beta$, put $r=1-\gamma$, $d=1+\delta$, $A=r({\delta }^2-\gamma )$ and $B=\gamma d^2$. At prescribed swimming speed $U$, eliminating wave speed using linear resistive-force propulsion of a travelling filament givesIts derivative has numerator $A{\beta }^2+2B\beta -B$. This numerator is strictly increasing on $(0,1)$, is negative at zero, and equals $(\delta +\gamma {)}^2$ at one. Thus its unique zero is the displayed global minimum. When $A=0$ the formula gives ${\beta }_{\ast }=1/2$; when $\delta =0$ it gives $1/(1+\sqrt{\gamma })$. A planar sawtooth realizes constant inclination in the ideal model; smooth rounding and elastic costs change that idealization.

For the homogeneous linear ordinary differential equation ${\sum }_{j=0}^n{a}_j{y}^{(j)}=0$, substitute $y=e^{rx}$. Dividing by the nonzero exponential gives the polynomial equation $P(r)={\sum }_j{a}_j{r}^j=0$. A simple root gives an exponential solution; a root of multiplicity $m$ gives $x^k{e}^{rx}$ for $0\le k<m$. Real equations admit real sine and cosine combinations of conjugate roots. These characteristic roots determine growth, decay and oscillation of the linearized modes.

For a nonzero vector $v$ and nonzero linear functional $f$ vanishing on $v$, this invertible map fixes the hyperplane $\ker f$ pointwise and has determinant one. Its inverse is $I-vf$, since $(vf{)}^2=0$. A symplectic transvection is obtained by taking $f=cB(-,v)$ for a nondegenerate alternating bilinear form.

The rank of a linear map is the dimension of its image of a linear map. In finite-dimensional spaces it equals the rank of a matrix representing the map in any bases.

For endomorphisms $S,T$ of the same finite-dimensional vector space, $ST=I$ makes $S$ surjective and $T$ injective. The rank-nullity theorem makes both invertible, hence $TS=I$. The finite-dimensional and equal-domain assumptions matter; unilateral shift operators on an infinite-dimensional sequence space give one-sided inverses that are not two-sided.

For a linear map $\alpha :U\to V$ between finite-dimensional vector spaces, take $\beta \in \mathrm{Hom}(V,U)$. The vector subspace $M^l(\alpha )=\{\beta :\beta \alpha =0\}$ consists of the linear maps factoring uniquely through $V/\mathrm{im}\alpha$, whereas $M^r(\alpha )=\{\beta :\alpha \beta =0\}$ consists of those whose image of a linear map lies in $\ker \alpha$. ThusWriting $u=\dim U$, $v=\dim V$ and $r=\mathrm{rank}\alpha$, their dimensions are $(v-r)u$ and $(u-r)v$. Passing to the dual map reverses composition and interchanges the corresponding left and right conditions. The finite-dimensional hypothesis makes these dimension differences ordinary integer subtraction.

A finite vector family is minimally dependent if it is linearly dependent and every proper subfamily is linearly independent. For $n\ge 2$, the family $e_1,\dots ,e_{n-1},{\sum }_{j=1}^{n-1}{e}_j$ in ${\mathbb{R}}^n$ is an example. The sum supplies a nonzero dependence. If the sum vector is omitted, the remaining vectors are a coordinate basis; if $e_j$ is omitted, its coordinate in any proposed relation forces the coefficient of the sum vector to vanish, then all other coefficients vanish. Every smaller proper subfamily lies in one of these independent $(n-1)$-element subfamilies.

A family is linearly independent if and only if every vector in its linear span has unique coefficients as a linear combination of that family. For a short proof, subtract two representations to get ${\sum }_i(a_i-b_i)v_i=0$; linear independence forces each difference to vanish. Conversely, uniqueness applied to the zero vector forces every linear relation to use only zero coefficients. This explains the practical role of linear independence: a coordinate vector relative to an ordered basis is unambiguous.

For pressureless linear cosmological density perturbations in an Einstein-de Sitter universe, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. The growth equation is an Euler differential equation whose indicial equation is $(p-2/3)(p+1)=0$. Hence the independent modes are ${\delta }_{+}\propto t^{2/3}\propto a$ and ${\delta }_{-}\propto t^{-1}\propto a^{-3/2}$. The first is the growing linear growth factor, while the second is a decaying transient.

For positive-definite within-group scatter $W$ and between-group scatter $B$, directions maximizing $a^{T}Ba/(a^{T}Wa)$ solve the displayed generalized eigenvalue problem. Put $u=W^{1/2}a$; the quotient becomes the Rayleigh quotient of $W^{-1/2}B{W}^{-1/2}$. Its ordered orthonormal eigenvectors give directions orthogonal in the $W$ inner product. There are at most $g-1$ nonzero directions, where $g$ is the number of groups. These directions summarize separation for linear discriminant analysis.

In a fixed frame, a first-order linear differential equation is Fuchsian at a finite point when its coefficient matrix has at most a simple pole there. Its residue is $R$. The global object is the matrix-valued one-form $A(z)dz$: in the coordinate $t=1/z$ its coefficient is $-t^{-2}A(1/t)$. A Fuchsian system has intrinsic regular singular points, but an intrinsically regular singular system may have higher-order coefficient poles in a different meromorphic frame.

A linear congruential generator has period $M$ for every seed if and only if $\gcd (c,M)=1$, every prime dividing $M$ divides $a-1$, and $4\mid M$ implies $4\mid a-1$. The coprimality of $c$ prevents the orbit from being trapped in a residue class; the multiplier conditions permit full-period lifting through each prime power. Combining the prime-power periods by the Chinese remainder theorem gives period $M$. For $M=12^k=2^{2k}{3}^k$, the multiplier criterion reduces to $a\equiv 1(\mathrm{mod}12)$, together with $\gcd (c,6)=1$. A positive increment alone is insufficient.

Birefringence is the existence of different optical propagation constants for different eigenpolarizations in an anisotropic material. It provides the propagation-mode separation needed for adiabatic optical following in a twisted nematic.

Two different refractive indices can produce two displaced images, as in the calcite double-image experiment, or a phase retardation between overlapping components, as in a waveplate. Spatial separation concerns where energy flux travels; phase retardation concerns the relative wave phase. A specimen need not show two visibly separated images to have birefringence.

An element $u$ of a ring is unipotent when $u-1$ is nilpotent. For a matrix, this says that every eigenvalue is one. A unipotent algebraic group consists of unipotent elements in a faithful linear representation.

A reductive algebraic group is a smooth connected affine algebraic group whose largest connected normal unipotent algebraic group is trivial.

A diagonalizable algebraic group is isomorphic to a closed subgroup of a product of copies of ${\mathbb{G}}_m$. Every rational representation of such a group decomposes into weight spaces.