The relaxation modulus is the shear-stress response at lag s to a unit small step of shear strain. By linear superposition it is the causal memory kernel multiplying shear rate. Its decaying part represents stored stress that relaxes after deformation ceases.

For the convention $e^{i\omega t}$, complex viscosity is the ratio of oscillatory shear stress amplitude to shear rate amplitude. A causal relaxation modulus gives the displayed transform. Its associated complex shear modulus is $G^{\ast }=i\omega {\eta }^{\ast }=G^{\prime }+i{G}^{\prime \prime }$, where $G^{\prime }$ is the storage modulus and $G^{\prime \prime }$ the loss modulus.

A small step of shear stress $s_{0}H(t)$ produces shear strain $\gamma (t)=s_{0}J(t)$ from a relaxed state. This time-dependent deformation under held stress is creep. The causal constitutive convolution gives $\hat{s}=p\hat{G}\hat{\gamma }$, proving the displayed Laplace transform relation. For a Maxwell fluid, $J(t)=1/G_0+t/(G_0\tau )$: an elastic jump followed by viscous flow.

For $x^{\prime }-Ax=e^{\lambda t}v$, substituting an exponential particular solution reduces the differential equation to $(\lambda I-A)u=v$. The displayed solution exists whenever $\lambda$ is not an eigenvalue of $A$. Resonance can require polynomial factors in time. The general solution adds $e^{tA}c$, expressed using the matrix exponential.

Minimizing $\Vert y-X\theta {\Vert }^2$ gives the displayed equations by differentiation. Equivalently, the regression residual $r=y-X\hat{\theta }$ satisfies $X^{T}r=0$: it is orthogonal to every column of the design matrix. If $X$ has full column rank, the minimizer is unique and the fitted values are the orthogonal projection of $y$ onto its column space. A rank-deficient design matrix can give multiple coefficient vectors, while the projected fitted vector is still unique.

For an intercept and one predictor, put $S_{xx}={\sum }_i(x_i-\bar{x}{)}^2$ and $S_{xy}={\sum }_i(x_i-\bar{x})(y_i-\bar{y})$. The equations give $\hat{\beta }=S_{xy}/S_{xx}$ and $\hat{\alpha }=\bar{y}-\hat{\beta }\bar{x}$, provided $S_{xx}>0$. In the five-point example in linear regression, $\bar{x}=2$, $\bar{y}=2.8$, $S_{xx}=10$ and $S_{xy}=10$, giving $\hat{\beta }=1$ and $\hat{\alpha }=0.8$.

The normal equations describe the mathematical optimum, but explicitly forming and inverting $X^{T}X$ can amplify numerical problems. A QR decomposition computes the same least-squares solution through orthogonal coordinates; a singular value decomposition also exposes nearly dependent columns. This connects statistical fitting to numerical linear algebra and the choice of a stable algorithm. ^[ref]

Multiple linear regression models a scalar response by an intercept and several predictors. It is linear in the regression coefficients, even when predictors include transformed variables or interactions. In an observational study, adjustment can address measured confounding, but does not automatically identify a causal effect. Improvements in fitted coefficient of determination require external predictive assessment rather than being treated as proof of improved future prediction.

For a square invertible constraint matrix $A$ in $\max \{c^{T}x:Ax\le b,x\ge 0\}$, suppose $y>0$ and $A^{T}y=c$. This fixed dual vector stays optimal under a changed right-hand side $b^{\prime }$ exactly when $A^{-1}{b}^{\prime }\ge 0$. Sufficiency follows from primal feasibility and equal primal/dual objectives. Necessity follows from complementary slackness: since every $y_i$ is positive, every optimal primal constraint must bind, forcing $x=A^{-1}{b}^{\prime }$. The inequalities define the whole validity cone, including degenerate boundary optima.

The commutator measures failure of two operators to commute, on a common domain where the products are defined. Direct expansion gives $[AB,C]=A[B,C]+[A,C]B$. A common eigenvector of $A,B$ lies in the kernel of $[A,B]$, so an injective commutator excludes all nonzero common eigenvectors. A nonzero commutator need not itself be injective, and therefore does not alone exclude an individual common eigenvector.

A linear operator $T$ is locally nilpotent if for each vector $v$ there is an integer $m(v)$ such that $T^{m(v)}v=0$. In finite dimension this is equivalent to being a nilpotent operator. Differentiation on the polynomial ring is locally nilpotent in characteristic zero but has no uniform nilpotence exponent.

A linear operator is semisimple if it becomes diagonalizable over an algebraic closure; equivalently, its minimal polynomial has no repeated root over that closure. If an invertible operator $A$ has finite order $r$ prime to the characteristic of a field, its minimal polynomial divides $X^r-1$. The derivative $r{X}^{r-1}$ is coprime to this polynomial, so $A$ is semisimple. Its invariant irreducible summands over a finite field become multiplication by scalars in suitable finite field extensions.

For a quasistatic beam made of a two-element series Maxwell material, $\dot{M}+M/\tau =EI{\partial }_t{y}_{xx}$. A force step therefore produces an instantaneous elastic deflection followed by unbounded linear creep, as displayed. It cannot have the finite long-time plateau of a Kelvin-Voigt cantilever creep response; a solid with finite relaxed modulus requires an additional elastic element.

For a half-space initially at rest and a wall given a step velocity $U$, momentum conservation and the Maxwell fluid stress law give $\tau u_{tt}+u_t=(G_0\tau /\rho )u_{yy}$. The displayed decaying Laplace transform solution obeys the wall condition. Inverting on a right-half-plane Bromwich contour yields a shear signal of speed $\sqrt{G_0/\rho }$. The limit $\tau \to 0$ at fixed $G_0\tau =\eta$ gives viscous error-function penetration; $\tau \to \infty$ at fixed $G_0$ gives a sharp elastic front.

For a no-slip channel with half-width $h$, a harmonic pressure gradient $\Delta p{e}^{i\omega t}$ and complex viscosity $\hat{\mu }=G_0\tau /(1+i\omega \tau )$, put $z^2=i\omega \rho h^2/\hat{\mu }$. Solving $\hat{\mu }{U}^{\prime \prime }-i\omega \rho U=\Delta p$ gives $U=-\Delta p[1-\cosh (zy/h)/\cosh z]/(i\omega \rho )$; integration gives the displayed flux per unit width. At low frequency it approaches $-2\Delta p{h}^3/(3G_0\tau )$. At fixed positive $h,\tau ,G_0,\rho$, the high-frequency leading flux is $-2\Delta ph/(i\omega \rho )$, although shear waves can persist in the local profile. If instead $\omega \tau \gg 1$ while $\rho {\omega }^2{h}^2/G_0\ll 1$, the negligible-inertia elastic limit is $Q\simeq -2i\omega \Delta p{h}^3/(3G_0)$. The last two limits impose different scale orderings.

For dust, curvature and a constant cosmological constant, write $H^2=A{a}^{-3}+B{a}^{-2}+C$. Then $\dot{H}=-3A{a}^{-3}/2-B{a}^{-2}$ and $\ddot{H}=H(9A{a}^{-3}/2+2B{a}^{-2})$, which imply $\ddot{H}+2H\dot{H}-3A{a}^{-3}H/2=0$. Thus $H$ is a solution of the pressureless linear matter perturbation growth equation. In an Einstein-de Sitter universe, $H\propto t^{-1}$ is the decaying mode and the independent growing mode is $t^{2/3}$. Additional dynamical components need not preserve this identity.

Use a monotone linear growth factor $b(t)$ as time, $\mathbf{u}=\dot{\mathbf{x}}/\dot{b}$, and $\psi =a\phi /(Cb)$ with $C=3H_0^2{\Omega }_{m0}/2$ and $a_0=1$. The cosmological Poisson equation becomes ${\nabla }^2\psi =\delta /b$; the continuity equation becomes ${\partial }_b\delta +\nabla \cdot [(1+\delta )\mathbf{u}]=0$. The material derivative is $D_b={\partial }_b+\mathbf{u}\cdot \nabla$. Substitution into the pressureless cosmological Euler equation, followed by the growth equation and $\dot{b}=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.

For an absolutely summable linear filter of a stationary time series, the filter gain is the modulus of its frequency response $a(\lambda )={\sum }_r{a}_r{e}^{ir\lambda }$. It multiplies a sinusoidal amplitude, while its square multiplies the spectral density of a stationary process. A zero removes the corresponding frequency; the filter gain of a composition is the product of the two filter gains.

Two absolutely summable linear filters of a stationary time series compose by convolution: $c_r={\sum }_j{b}_j{a}_{r-j}$ and ${\sum }_r|c_r|\le ({\sum }_r|a_r|)({\sum }_j|b_j|)$. Absolute convergence in the space of square-integrable random variables justifies exchanging the sums. The frequency responses multiply, giving the spectral density transformation under a linear filter twice.

For absolutely summable linear-filter coefficients, the generating function is the displayed Laurent series, absolutely convergent on the unit circle. If the filter is one-sided, it is a power series analytic inside the unit disk. A two-sided filter need not have a nonempty larger annulus of convergence. Substituting the backshift operator gives the formal notation $Y=A(B)X$.

A linear relation among vectors $v_1,\dots ,v_n$ is a coefficient tuple $(a_1,\dots ,a_n)$ satisfying ${\sum }_i{a}_i{v}_i=0$. It is nontrivial if at least one coefficient is nonzero. For the matrix with these vectors as columns, the space of relations is its kernel. The vectors are linearly independent exactly when this kernel is zero. An invertible change of coordinates preserves every relation, since $S{\sum }_i{a}_i{v}_i=0$ if and only if ${\sum }_i{a}_i{v}_i=0$.

Consider the vectors $1,i$ in the complex numbers, where $i$ is the imaginary unit. With scalar field $\mathbb{R}$, a linear relation $a\cdot 1+b\cdot i=0$ has $a,b\in \mathbb{R}$. Taking its real part and imaginary part gives $a=b=0$, so the pair is linearly independent. Every complex number $x+iy$ also has coefficients $(x,y)$ in the real numbers, so $1,i$ form a basis and the dimension of a vector space is two.

With scalar field $\mathbb{C}$, the same pair is linearly dependent, sinceuses the nonzero coefficients $(i,-1)$ from the complex numbers. The single vector $1$ is now a basis, because $z=z\cdot 1$ for every complex number $z$, giving dimension of a vector space one. For example $3+i=3\cdot 1+1\cdot i=(3+i)\cdot 1+0\cdot i$: the second representation uses a coefficient unavailable over the real numbers. Specifying the field is part of the definition of linear independence.