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If the columns $v_1,\dots ,v_m$ of an $n$ by $m$ matrix are independent, some $m$ by $m$ row minor is nonzero. A nonzero term of that minor's determinant selects distinct nonzero coordinates $f(i)$, and replacing the corresponding standard basis vectors $e_{f(i)}$ by the $v_i$ leaves a basis.

An infinite locally finite connected graph is a transient graph exactly when it supports a unit flow from a vertex to infinity with finite energy of a flow.

For every null vector $k$, put $q^{\rho }=k_{\mu }{F}^{\mu \rho }$. Antisymmetry gives $q\cdot k=0$, so $q$ has nonnegative norm and

A null field has both electromagnetic invariants zero:For a nonzero constant null field, the mixed tensor is nilpotent of index three,

Phase matching fixes a common frequency and tangential wavenumber for every incident, reflected, and transmitted elastic mode.

An obliquely incident in-plane P-wave generally produces reflected and transmitted P and SV waves; SH polarization decouples.

For $d{s}^2=-d{t}^2+a(t{)}^{2}d{\mathbf{x}}^2$, the nonzero connection coefficients areWith $G_t^t=-3H^2$ and
$G_i^j=-(2\ddot{a}/a+H^2){\delta }_i^j$, direct substitution into the mixed-tensor covariant derivative gives zero.

When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.

A left eigenvector satisfies $u^{T}A=\lambda u^T$, equivalently $A^{T}u=\lambda u$ over the real numbers.

A right eigenvector of a square matrix $A$ is a nonzero column vector $v$ satisfying $Av=\lambda v$ for some scalar $\lambda$. The scalar is its eigenvalue. Requiring $v\ne 0$ is essential, since the zero vector satisfies the displayed equation for every $\lambda$.

For example,Thus the two displayed vectors are right eigenvectors with different eigenvalues. A left eigenvector instead acts on $A$ from the left; specifying the side matters for a general matrix.

An eigenbasis is a basis consisting of eigenvectors of one operator.

The binary entropy function isIt is concave on $[0,1]$ and satisfies $H_2(x)\ge 2x$ for $0\le x\le 1/2$ by comparison with the chord from $(0,0)$ to $(1/2,1)$.

Every Dynkin system containing a pi-system $\mathcal{P}$ also contains the sigma-algebra $\sigma (\mathcal{P})$. Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.

The dust transparency radius is where a swept shell's ultraviolet optical depth equals one. For a gas fraction $f_g$ in a singular isothermal sphere, $R_{\tau }={\kappa }_{\mathrm{UV}}{f}_g{\sigma }^2/(2\pi G)$.

On the real line, initial data $\delta (x-2\sqrt{D})$ and a source $-A\delta (x+2\sqrt{D})\delta (t-1)$ produceAt $x=0,t=2$, the two terms cancel exactly for $A=\sqrt{e/2}$.

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.

A discrete valuation on a field $K$ is a surjective group homomorphism $v:K^{\times }\to \mathbb{Z}$ satisfying $v(x+y)\ge \min (v(x),v(y))$ whenever $x+y\ne 0$. Its valuation ring is $\{0\}\cup \{x:v(x)\ge 0\}$.

Every principal ideal $I$ of a number field has a generator $\alpha$ satisfyingat every archimedean embedding. Multiply an arbitrary generator by a unit so that its logarithmic unit-lattice coordinates lie in a fixed bounded fundamental parallelepiped.

The Relaxation time of a finite reversible lazy Markov chain is the reciprocal of its spectral gap: $t_{\mathrm{rel}}=(1-{\lambda }_2{)}^{-1}$.