Dictionary

$A^{\mathsf T}Ax = O$ exactly when $Ax = O$.

Vol. 2, 2.10 Least squares and regression as projection

A set $A$ is closed if and only if every limit point of $A$ belongs to $A$.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

If $Z = P \cup Q$ is a separation and $C \subseteq Z$ is connected, then $C \subseteq P$ or $C \subseteq Q$.

Vol. 2, 4.4 Connectedness and path-connectedness

If $K$ is compact and $f : K \to Y$ is a continuous bijection onto a metric space, then $f^{-1}$ is continuous, so $f$ is a homeomorphism.

Vol. 2, 4.5 Continuity between metric spaces

If $h$ is continuous on an open set and $\iiint_Eh\,dV = 0$ for every ball $E$ inside it, then $h = 0$ everywhere.

Vol. 2, 6.8 Maxwell's equations, conservation laws and heat

If $x \mapsto L(x, \lambda) = f(x) - \sum\lambda_i(g_i(x) - c_i)$ is convex and $a$ satisfies $\nabla f(a) = \sum\lambda_i\nabla g_i(a)$ and $g(a) = c$, then $a$ is an absolute minimum of $f$ on $\{g = c\}$.

Vol. 2, 5.8 Constrained extrema and Lagrange multipliers

$F(x) \to b$ as $x \to a$ if and only if $F_k(x) \to b_k$ for every component $k$.

Vol. 2, 5.1 Scalar and vector fields, limits and continuity

In a field, ab = 0 implies a = 0 or b = 0.

Vol. 2, 3.9 Groups, rings and fields: a preview built from what you know

For $f: A \to B$ with $a \sim b$ iff $f(a) = f(b)$, the rule $[a] \mapsto f(a)$ is a well-defined bijection from $A / {\sim}$ onto the range $f(A)$.

Vol. 2, 0.4 Equivalence relations and partitions

Put $x = (y, y', \dots, y^{(n-1)})$. Then $y^{(n)} = f(t, y, \dots, y^{(n-1)})$ is the same as $x' = F(t, x)$ for a vector function $F$.

Vol. 2, 7.1 Existence and uniqueness for differential equations

A function has at most one left inverse, and a left inverse is also a right inverse.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

If $A$ and $B$ are $n \times n$ and $BA = I$, then $AB = I$ too, so $B = A^{-1}$.

Vol. 2, 2.6 Systems of equations and elimination

If $F(x) \to b$ as $x \to a$ and a path $r(t) \to a$ with $r(t) \ne a$, then $F(r(t)) \to b$ along the path.

Vol. 2, 5.1 Scalar and vector fields, limits and continuity

$L(f + g) = Lf + Lg$ and $L(cf) = c\,Lf$, so the solutions of $Ly = 0$ form the null space of a linear map.

Vol. 2, 7.2 Linear differential equations

If a $C^2$ field has a local minimum at a critical point $a$, then every eigenvalue of $H_f(a)$ is at least $0$; if it has a local maximum, every eigenvalue is at most $0$.

Vol. 2, 5.6 Taylor's formula, the Hessian and extrema

For a square matrix, $\det A \ne 0$ if and only if $A$ is invertible, if and only if the rows are independent, if and only if the columns are independent.

Vol. 2, 2.7 Determinants

If $W(f_1, \dots, f_n)(t_0) \ne 0$ at even one point $t_0$ of $J$, then $f_1, \dots, f_n$ are linearly independent on $J$.

Vol. 2, 7.3 The Wronskian, variation of parameters and undetermined coefficients

If $\lVert \cdot \rVert$ is a norm on $V$, then $d(x, y) = \lVert x - y \rVert$ is a metric on $V$.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

With no source, the stored energy $W = \frac12LI^2 + \frac{q^2}{2C}$ satisfies $\dot W = -RI^2 \le 0$. With a source, $\dot W = IE - RI^2$.

Vol. 2, 7.7 Applications to springs, circuits, epidemics and qubits

A non-zero polynomial of degree $n$ has at most $n$ roots (real or complex).

Vol. 2, 0.6 Complex numbers and polynomials, done properly

If $A$ is a real symmetric matrix with eigenvalues $\lambda_{\min} \le \dots \le \lambda_{\max}$, then $\lambda_{\min}\lVert v \rVert^2 \le v^{\mathsf T}Av \le \lambda_{\max}\lVert v \rVert^2$ for every $v$, with equality at the corresponding eigenvectors.

Vol. 2, 5.6 Taylor's formula, the Hessian and extrema

If g has finite order k then ⟨g⟩ = {e, g, g², …, g^(k−1)} has exactly k elements, and gᵐ = e exactly when k divides m.

Vol. 2, 3.9 Groups, rings and fields: a preview built from what you know

If $P$ is stochastic and $x$ is a probability vector, then $Px$ is a probability vector. Also a product of stochastic matrices is stochastic.

Vol. 2, 2.11 Markov chains and PageRank

 

Vol. 2, 5.1 Scalar and vector fields, limits and continuity

No solution if $b$ is not in the range of $A$. Otherwise one solution if the null space is $\{O\}$, and infinitely many if it is bigger.

Vol. 2, 2.6 Systems of equations and elimination

$F = (F_1, \dots, F_m)$ is differentiable at $a$ if and only if every $F_i$ is, and then $F'(a)(y) = (\nabla F_1(a)\cdot y, \dots, \nabla F_m(a)\cdot y)$, whose matrix is the Jacobian.

Vol. 2, 5.3 The total derivative, the Jacobian matrix and tangent planes

If $b_1, \dots, b_n$ solve $y^{(n)} + a_{n-1}y^{(n-1)} + \dots = 0$, their Wronskian satisfies $W' = -a_{n-1}W$, so $W(t) = W(t_0)\exp\!\big(-\int_{t_0}^{t}a_{n-1}\big)$.

Vol. 2, 7.3 The Wronskian, variation of parameters and undetermined coefficients

A group in which ab = ba for all a and b.

Vol. 2, 3.9 Groups, rings and fields: a preview built from what you know

$\sum a_k$ converges absolutely if $\sum |a_k|$ converges.

Vol. 2, 1.6 Series, power series, and uniform convergence

If $\sum |a_k|$ converges, then $\sum a_k$ converges, and $\left|\sum a_k\right| \le \sum |a_k|$.

Vol. 2, 1.6 Series, power series, and uniform convergence

A point $a$ of the domain $S$ with $f(x) \le f(a)$ for every $x \in S$ (a global maximum).

Vol. 2, 5.6 Taylor's formula, the Hessian and extrema

A point $a$ of the domain $S$ with $f(x) \ge f(a)$ for every $x \in S$ (a global minimum).

Vol. 2, 5.6 Taylor's formula, the Hessian and extrema

If $\sum_k\lVert M_k\rVert$ converges as a series of numbers, then $\sum_kM_k$ converges in the space of matrices.

Vol. 2, 7.4 Systems of equations and the matrix exponential

If $f$ is finitely additive on a Boolean algebra, then $f(A \cup B) = f(A) + f(B \setminus A)$ and $f(A \cup B) = f(A) + f(B) - f(A \cap B)$ for all $A, B$ in the algebra.

Vol. 2, 10.1 Probability as measure

If $a < c < b$ and $f$ is integrable on $[a, c]$ and on $[c, b]$, then it is integrable on $[a, b]$ and $\int_a^b f = \int_a^c f + \int_c^b f$.

Vol. 2, 1.5 The Riemann integral: area as a limit

The adjoint $A^*$ is the map with $(Ax, y) = (x, A^*y)$ for all $x, y$; for real matrices with the dot product it is the transpose.

Vol. 2, 3.1 Symmetric matrices and the spectral theorem

If $a_n \to a$ and $b_n \to b$, then $a_n + b_n \to a + b$, $c\,a_n \to ca$, $a_n b_n \to ab$, and $a_n/b_n \to a/b$ when $b \ne 0$.

Vol. 2, 1.2 Sequences and limits, from zero

If $V$ has a basis with $n$ elements, every basis of $V$ has $n$ elements.

Vol. 2, 2.2 Span, independence, basis, dimension

For any norm $\lVert \cdot \rVert$ on $\mathbb R^n$ there are constants $0 < m \le M$ with $m\lVert x \rVert_1 \le \lVert x \rVert \le M\lVert x \rVert_1$ for all $x$.

Vol. 2, 4.5 Continuity between metric spaces

If $b_k$ decreases to $0$, then $\sum (-1)^{k+1} b_k$ converges, and the error after $n$ terms is at most $b_{n+1}$.

Vol. 2, 1.6 Series, power series, and uniform convergence

For the magnetic field of steady currents, the circulation of $B$ round a closed loop $C$ is $\mu_0$ times the current through any surface bounded by $C$: $\oint_CB\cdot d\ell = \mu_0\iint_SJ\cdot n\,dS$, locally $\nabla\times B = \mu_0J$.

Vol. 2, 6.8 Maxwell's equations, conservation laws and heat

If $T(x) = \lambda x$ and $T(x) = \mu x$ with $x \ne O$, then $\lambda = \mu$.

Vol. 2, 2.8 Eigenvalues and eigenvectors

$P \Rightarrow Q$ and $\neg Q \Rightarrow \neg P$ are true in exactly the same situations.

Vol. 2, 0.1 How to read a theorem

A continuous function $y$ solves $y' = f(t, y)$, $y(t_0) = y_0$ on an interval exactly when $y(t) = y_0 + \int_{t_0}^{t} f(s, y(s))\,ds$ for every $t$ in it.

Vol. 2, 7.1 Existence and uniqueness for differential equations

A function $f$ is analytic on $(x_0 - r, x_0 + r)$ if it equals a power series $\sum a_n(x - x_0)^n$ at every point of that interval.

Vol. 2, 7.5 Power series solutions, Legendre and Bessel

A constant-coefficient operator $B$ with $Bf = 0$: it sends the function $f$ to zero. Example: $D - r$ annihilates $e^{rt}$, $(D - r)^m$ annihilates $t^{m-1}e^{rt}$.

Vol. 2, 7.3 The Wronskian, variation of parameters and undetermined coefficients

For a subspace U of V, the set U⁰ of functionals that are 0 on every vector of U.

Vol. 2, 3.7 Dual spaces and quotient spaces

 

Vol. 2, 1.1 The real numbers and completeness

 

Vol. 2, 1.1 The real numbers and completeness

For a positively oriented piecewise smooth Jordan curve $\alpha(t) = (X(t), Y(t))$ bounding $S$, and any real $c$, $\text{area}(S) = \int_a^b\big[(c - 1)YX' + cXY'\big]\,dt$; with $c = 1, 0, \frac12$: $\text{area}(S) = \oint x\,dy = -\oint y\,dx = \frac12\oint(x\,dy - y\,dx)$.

Vol. 2, 6.4 Green's theorem and change of variables

 

Vol. 2, 6.4 Green's theorem and change of variables

If $a \equiv a'$ and $b \equiv b'$ (mod $n$), then $a + b \equiv a' + b'$ and $ab \equiv a'b'$; so addition and multiplication of classes are well defined.

Vol. 2, 0.5 Integers mod n, gcd and Bézout

 

Vol. 2, 7.6 Phase portraits, stability and linearization

The matrix $[A \mid b]$: the coefficient matrix with the right-hand side added as an extra last column.

Vol. 2, 2.6 Systems of equations and elimination

A system $x' = F(x)$ whose right-hand side does not depend on the time $t$.

Vol. 2, 7.6 Phase portraits, stability and linearization

If $d$ is a determinant function and $F$ satisfies only Axioms 1 to 3, then $F(A) = d(A)\,F(I)$ for every $A$.

Vol. 2, 2.7 Determinants

Compute the derivative of a scalar output with respect to every input by applying the chain rule backwards through the computational graph, reusing each intermediate result once.

Vol. 2, 3.8 Matrix calculus and backpropagation

A contraction $T$ of a non-empty complete metric space has exactly one fixed point $x^*$; from any start $x_0$ the iterates $x_{n+1} = T x_n$ converge to $x^*$, with $d(x_n, x^*) \le \dfrac{q^n}{1 - q}\,d(x_1, x_0)$.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

$R_0 = \beta/\gamma$: the average number of people infected by one infected person during the whole infectious period, when everyone else is susceptible.

Vol. 2, 7.7 Applications to springs, circuits, epidemics and qubits

From the three axioms: $P(\varnothing) = 0$; probability is finitely additive; $P(A^c) = 1 - P(A)$; $A \subseteq B$ implies $P(A) \le P(B)$; every $P(A)$ lies in $[0, 1]$; and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Vol. 2, 10.1 Probability as measure

An independent set that spans the whole space: enough vectors to build everything, and none wasted.

Vol. 2, 2.2 Span, independence, basis, dimension

The $m$ functions $e^{rt}, te^{rt}, \dots, t^{m-1}e^{rt}$ are a basis of the solutions of $(D - r)^my = 0$.

Vol. 2, 7.2 Linear differential equations

If $P_A(t) = \prod_{i=1}^{k}(t - r_i)^{m_i}$ with distinct roots $r_i$, the $n = \sum m_i$ functions $t^je^{r_it}$ ($i \le k$, $0 \le j < m_i$) are a basis of the solutions of $Ay = 0$.

Vol. 2, 7.2 Linear differential equations

 

Vol. 2, 10.1 Probability as measure

$J_n(x) = \sum_{k \ge 0}\dfrac{(-1)^k}{k!\,(k+n)!}\left(\dfrac x2\right)^{2k+n}$, the solution of Bessel's equation $x^2y'' + xy' + (x^2 - n^2)y = 0$ that is finite at $x = 0$ (for $n = 0, 1, 2, \dots$).

Vol. 2, 7.5 Power series solutions, Legendre and Bessel

Among all points of $S$, the projection $s$ is the unique closest point to $x$: $\lVert x - s\rVert \le \lVert x - t\rVert$ for every $t \in S$.

Vol. 2, 2.9 Inner products, orthogonality, Gram–Schmidt, projection

The gcd of $a$ and $b$ is always an integer combination of them: $\gcd(a, b) = ax + by$ for some integers $x, y$. In fact it is the smallest positive number of that form.

Vol. 2, 0.5 Integers mod n, gcd and Bézout

$f: A \to B$ is bijective exactly when there is a function $g: B \to A$ that undoes it in both orders; this $g$ is the inverse $f^{-1}$.

Vol. 2, 0.3 Sets, functions, and counting infinity

$f$ is bijective if it is both injective and surjective: every element of the codomain is hit by exactly one input.

Vol. 2, 0.3 Sets, functions, and counting infinity

Every closed bounded subset of $\mathbb R^n$ is sequentially compact: each sequence in it has a subsequence converging to a point of it.

Vol. 2, 4.3 Compactness

Every bounded sequence has a convergent subsequence.

Vol. 2, 1.2 Sequences and limits, from zero

A non-empty family of subsets of a set $S$ that contains the union $A \cup B$ and the complement $A^c = S \setminus A$ whenever it contains $A$ and $B$.

Vol. 2, 10.1 Probability as measure

For $f(x, y)$ with one constraint $g$ at a regular point: $\det B \gt 0$ means a strict local constrained maximum, $\det B \lt 0$ a strict local constrained minimum, where $B$ is the $3 \times 3$ bordered Hessian.

Vol. 2, 5.8 Constrained extrema and Lagrange multipliers

The $(n + m) \times (n + m)$ symmetric matrix $B = \begin{bmatrix}0 & Dg(a)\\ Dg(a)^{\mathsf T} & H\end{bmatrix}$, where $H = D^2f(a) - \sum\lambda_iD^2g_i(a)$ is the Hessian in $x$ of the Lagrangian.

Vol. 2, 5.8 Constrained extrema and Lagrange multipliers

The boundary of $A$ is every point whose every ball touches both $A$ and the complement of $A$: the edge of $A$.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

A set is bounded above if it has an upper bound, bounded below if it has a lower bound, and bounded if both.

Vol. 2, 1.1 The real numbers and completeness

A set $K$ is bounded if it fits inside some ball: there are a point $p$ and a radius $R$ with $K \subseteq B(p, R)$.

Vol. 2, 4.3 Compactness

If $f$ is bounded on a rectangle $Q$ and the set of points where $f$ is not continuous has content zero, then $f$ is integrable on $Q$.

Vol. 2, 6.3 Multiple integrals

If $T(e_k) = u_k$ on a basis, then $T$ is onto $W$ exactly when the $u_k$ span $W$, one-to-one exactly when they are independent, and an isomorphism exactly when they are a basis of $W$.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

Sums, scalar multiples, dot products, norms and compositions of continuous fields are continuous; so are the coordinate functions, linear maps, polynomials, and quotients where the denominator is not zero.

Vol. 2, 5.1 Scalar and vector fields, limits and continuity

Compositions of continuous maps are continuous; sums, multiples, products and quotients (with non-zero denominator) of continuous real functions are continuous; a map into $\mathbb R^m$ is continuous iff each of its $m$ coordinate functions is; coordinate projections are continuous.

Vol. 2, 4.5 Continuity between metric spaces

Every Cauchy sequence of continuous functions on $[a, b]$, in the sup metric, converges uniformly to a continuous function.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

The loss of most or all correct digits when two nearly equal numbers, each carrying a small rounding error, are subtracted.

Vol. 2, 15.1 Numerical analysis

A sequence of real numbers converges if and only if it is Cauchy.

Vol. 2, 1.2 Sequences and limits, from zero

 

Vol. 2, 1.6 Series, power series, and uniform convergence

 

Vol. 2, 1.2 Sequences and limits, from zero

 

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

$|(x, y)| \le \lVert x\rVert\,\lVert y\rVert$, with equality exactly when $x$ and $y$ are linearly dependent.

Vol. 2, 2.9 Inner products, orthogonality, Gram–Schmidt, projection

If $g, k$ are continuous on $[0, h]$, differentiable on $(0, h)$ and $k' \ne 0$ there, then $\dfrac{g(h) - g(0)}{k(h) - k(0)} = \dfrac{g'(\tau)}{k'(\tau)}$ for some $\tau \in (0, h)$.

Vol. 2, 15.1 Numerical analysis

Every square matrix satisfies its own characteristic polynomial: if p(λ) = det(λI − A), then p(A) = 0.

Vol. 2, 3.6 Cayley–Hamilton, Schur, and the Jordan form

The origin of $x' = Ax$ when $A$ has purely imaginary eigenvalues $\pm ib$, $b \ne 0$: every orbit is a closed curve (an ellipse) around it.

Vol. 2, 7.6 Phase portraits, stability and linearization

For a wire with density $\varphi$, the mass is $M = \int_C \varphi\,ds$, and the centre of mass is the point $(\bar x, \bar y, \bar z)$ with $\bar x\,M = \int_C x\,\varphi\,ds$, and likewise for $y$ and $z$.

Vol. 2, 6.1 Paths and line integrals

If $g$ is differentiable at $a$ and $f$ at $g(a)$, then $(f \circ g)'(a) = f'(g(a))\,g'(a)$.

Vol. 2, 1.4 The derivative as best linear approximation

The matrix $C$ whose column $j$ is the new basis vector $f_j$ written in old coordinates; it converts new coordinates into old ones.

Vol. 2, 2.5 Matrices as maps, multiplication, change of basis

If $A$ and $B$ are the matrices of $T$ in the old and new bases, then $B = C^{-1}AC$. Conversely every $C^{-1}AC$ is the matrix of the same map in some basis.

Vol. 2, 2.5 Matrices as maps, multiplication, change of basis

If $g: T \to S$ is one-to-one and $C^1$ with $J \ne 0$ (allowed to fail on a set of content zero) and $f$ is continuous on $S$, then $\iint_Sf\,dx\,dy = \iint_Tf(X(u, v), Y(u, v))\,|J(u, v)|\,du\,dv$.

Vol. 2, 6.4 Green's theorem and change of variables

A curve $(x(s), y(s))$ along which the PDE $a\,u_x + b\,u_y + c\,u = f$ turns into an ordinary differential equation: $x' = a$, $y' = b$.

Vol. 2, 5.9 Partial differential equations

The polynomial $p_A(\lambda) = \det(\lambda I - A)$. Its roots are exactly the eigenvalues of $A$.

Vol. 2, 2.8 Eigenvalues and eigenvectors

For $A = D^n + a_{n-1}D^{n-1} + \dots + a_0$, the polynomial $P_A(t) = t^n + a_{n-1}t^{n-1} + \dots + a_0$. Its roots decide the solutions.

Vol. 2, 7.2 Linear differential equations

The function $\rho(r)$ such that the charge in a region $E$ is $Q = \iiint_E\rho\,dV$.

Vol. 2, 6.8 Maxwell's equations, conservation laws and heat

The $n+1$ roots of $T_{n+1}$: $x_k = \cos\dfrac{(2k+1)\pi}{2n+2}$ for $k = 0, \dots, n$. They are the projections onto the $x$-axis of equally spaced points on a semicircle, so they crowd toward the ends.

Vol. 2, 15.1 Numerical analysis

$T_0 = 1$, $T_1 = x$, and $T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x)$. The key property is $T_n(\cos\theta) = \cos(n\theta)$.

Vol. 2, 15.1 Numerical analysis

 

Vol. 2, 10.1 Probability as measure

Among all monic polynomials of degree $m \ge 1$, none has a smaller largest absolute value on $[-1, 1]$ than $2^{1-m}$, and $T_m/2^{m-1}$ achieves it.

Vol. 2, 15.1 Numerical analysis

The circulation of a field $F$ around a closed curve $C$ is $\oint_C F \cdot dr$.

Vol. 2, 6.1 Paths and line integrals

A field is of class $C^k$ on an open set if all its partial derivatives of order at most $k$ exist and are continuous there; $C^\infty$ means $C^k$ for every $k$.

Vol. 2, 5.5 Higher partial derivatives and the equality of mixed partials

 

Vol. 2, 7.4 Systems of equations and the matrix exponential

 

Vol. 2, 7.6 Phase portraits, stability and linearization

A form $\omega$ with $d\omega = 0$.

Vol. 2, 6.7 Differential forms and the one big theorem

A path $r: [a, b] \to \mathbb R^n$ with $r(a) = r(b)$: it ends where it starts.

Vol. 2, 6.1 Paths and line integrals

A set is closed if its complement is open.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

A point lies in the closure of $A$ if and only if some sequence in $A$ converges to it. Hence $A$ is closed if and only if the limit of every convergent sequence of points of $A$ lies in $A$.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

If $Q$ is compact and $F \subseteq Q$ is closed, then $F$ is compact.

Vol. 2, 4.3 Compactness

Inside a complete space, a subset with the same distance is complete if and only if it is closed.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

A surface without a boundary curve of its own, such as the boundary of a solid region: a sphere, the surface of a cube or a torus. An open surface such as a bowl or a cube without a lid has a boundary curve.

Vol. 2, 6.6 Curl, divergence, Stokes' theorem and Gauss's theorem

A set is closed under an operation when applying it to elements of the set never produces something outside the set.

Vol. 2, 2.1 Vector spaces and subspaces

The closure of $A$ is every point whose every ball touches $A$: the set $A$ together with its edge.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

If $A$ is connected and $A \subseteq B \subseteq \overline A$, then $B$ is connected.

Vol. 2, 4.4 Connectedness and path-connectedness

Subsets of countable sets are countable; the image of a countable set under any function is countable; a product of two countable sets is countable.

Vol. 2, 0.3 Sets, functions, and counting infinity

The number $C_{kj} = (-1)^{k+j}\det A_{kj}$: the determinant of the minor with a checkerboard sign.

Vol. 2, 2.7 Determinants

$A\,(\operatorname{cof}A)^{\mathsf T} = (\det A)\,I$, so when $\det A \ne 0$, $A^{-1} = \frac1{\det A}(\operatorname{cof}A)^{\mathsf T}$.

Vol. 2, 2.7 Determinants

The matrix whose $(i, j)$ entry is the cofactor $C_{ij}$. Its transpose is the adjugate.

Vol. 2, 2.7 Determinants

The span of the columns of $A$. It equals the range of $x \mapsto Ax$.

Vol. 2, 2.6 Systems of equations and elimination

A metric space is compact if and only if it is complete and totally bounded.

Vol. 2, 4.3 Compactness

In a metric space, a set is compact if and only if it is sequentially compact.

Vol. 2, 4.3 Compactness

A set $K$ is compact if every open cover of $K$ has a finite subcover.

Vol. 2, 4.3 Compactness

In any metric space, every compact set is closed and bounded.

Vol. 2, 4.3 Compactness

The matrix $A(t)$ with ones directly above the diagonal and the last row $(-a_0, -a_1, \dots, -a_{n-1})$, for which $x = (y, y', \dots, y^{(n-1)})$ satisfies $x' = A(t)x + (0, \dots, 0, R)$.

Vol. 2, 7.2 Linear differential equations

For every $x \in \mathbb R^n$: $\lVert x \rVert_\infty \le \lVert x \rVert_2 \le \lVert x \rVert_1 \le \sqrt n\,\lVert x \rVert_2 \le n\,\lVert x \rVert_\infty$.

Vol. 2, 4.1 Metric and normed spaces; open and closed sets

If $0 \le a_k \le b_k$ and $\sum b_k$ converges, then $\sum a_k$ converges; if $\sum a_k$ diverges, so does $\sum b_k$.

Vol. 2, 1.6 Series, power series, and uniform convergence

$A^c$ is everything in the universe $U$ that is not in $A$.

Vol. 2, 0.3 Sets, functions, and counting infinity

If $A$ and $B$ are independent, so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$.

Vol. 2, 10.1 Probability as measure

A metric space in which every Cauchy sequence converges to a point of the space.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

Every non-empty set of real numbers that is bounded above has a supremum in $\mathbb R$.

Vol. 2, 1.1 The real numbers and completeness

An ordered pair $(a, b)$ of real numbers, added by $(a,b) + (c,d) = (a+c, b+d)$ and multiplied by $(a,b)(c,d) = (ac - bd,\ ad + bc)$.

Vol. 2, 0.6 Complex numbers and polynomials, done properly

If a polynomial with real coefficients has root $z$, then $\bar z$ is also a root.

Vol. 2, 0.6 Complex numbers and polynomials, done properly

With $N$ pieces of width $h$: $T_N = h\big(\tfrac12 f_0 + f_1 + \cdots + f_{N-1} + \tfrac12 f_N\big)$ has error at most $\dfrac{(b-a)h^2}{12}M_2$, and (for $N$ even) $S_N = \dfrac h3\big(f_0 + 4f_1 + 2f_2 + 4f_3 + \cdots + 4f_{N-1} + f_N\big)$ has error at most $\dfrac{(b-a)h^4}{180}M_4$.

Vol. 2, 15.1 Numerical analysis

The map $ST$ that does $T$ first and then $S$: $(ST)(x) = S(T(x))$.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

$(S + T)R = SR + TR$ and $R(S + T) = RS + RT$ (the second needs $R$ linear); scalars pass through too.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

For any three functions that can be chained, $R(ST) = (RS)T$.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

If $f$ and $g$ are injective, so is $g \circ f$; if both are surjective, so is $g \circ f$; hence a composition of bijections is a bijection.

Vol. 2, 0.3 Sets, functions, and counting infinity

If $T: U \to V$ and $S: V \to W$ are linear, then $ST: U \to W$ is linear.

Vol. 2, 2.4 Inverses, one-to-one maps, isomorphism

A diagram of a calculation: nodes are intermediate values, arrows say which values each one is computed from.

Vol. 2, 3.8 Matrix calculus and backpropagation

Row reduce $[A \mid I]$. If the left block becomes $I$, the right block is $A^{-1}$. If the left block cannot become $I$, $A$ has no inverse.

Vol. 2, 2.6 Systems of equations and elimination

$\kappa(A) = \lVert A \rVert\,\lVert A^{-1} \rVert = \sigma_1/\sigma_n$: the worst-case factor by which A magnifies relative errors in solving Ax = b.

Vol. 2, 3.5 Matrix norms, conditioning, and why numerics fail

 

Vol. 2, 10.1 Probability as measure

 

Vol. 2, 10.1 Probability as measure

$a \equiv b \pmod n$ means $n$ divides $a - b$: $a$ and $b$ leave the same remainder on division by $n$.

Vol. 2, 0.5 Integers mod n, gcd and Bézout

The conjugate of $z = a + bi$ is $\bar z = a - bi$, its mirror image in the real axis.

Vol. 2, 0.6 Complex numbers and polynomials, done properly

A space $A$ is connected if and only if the only subsets that are both open and closed in $A$ are $\varnothing$ and $A$.

Vol. 2, 4.4 Connectedness and path-connectedness

A set (a metric space) is connected if it has no separation: it cannot be written as two disjoint non-empty open pieces.

Vol. 2, 4.4 Connectedness and path-connectedness

A subset of $\mathbb R$ is connected if and only if it is an interval.

Vol. 2, 4.4 Connectedness and path-connectedness

In a finite group G: the order of every element divides |G|, so g^|G| = e; a group of prime order p is cyclic; and, for the group of invertible classes mod n, a^φ(n) ≡ 1 (Euler), in particular a^(p−1) ≡ 1 for prime p (Fermat).

Vol. 2, 3.9 Groups, rings and fields: a preview built from what you know

In any group: the identity is unique; each element has exactly one inverse; (ab)⁻¹ = b⁻¹a⁻¹; ab = ac implies b = c (and ba = ca implies b = c); and ax = b has the unique solution x = a⁻¹b.

Vol. 2, 3.9 Groups, rings and fields: a preview built from what you know

$\det(A^{-1}) = 1/\det A$, $\det(A^k) = (\det A)^k$, $\det(AB) = \det(BA)$, and $\det(cA) = c^n\det A$ for an $n \times n$ matrix.

Vol. 2, 2.7 Determinants

An operator $A = D^n + a_{n-1}D^{n-1} + \dots + a_0$ with numbers $a_k$ as coefficients. It equals $P_A(D)$ for the polynomial $P_A(t) = t^n + a_{n-1}t^{n-1} + \dots + a_0$.

Vol. 2, 7.2 Linear differential equations

If $A = P_A(D)$ and $B = P_B(D)$ then $AB = (P_AP_B)(D)$. In particular $AB = BA$.

Vol. 2, 7.2 Linear differential equations

A point $a$ with $g(a) = c$ such that $f(x) \le f(a)$ (a maximum) or $f(x) \ge f(a)$ (a minimum) for all nearby $x$ that also satisfy $g(x) = c$.

Vol. 2, 5.8 Constrained extrema and Lagrange multipliers

 

Vol. 2, 6.3 Multiple integrals

If a conserved quantity has density $u$ and flux density $j$, then $\partial_tu + \nabla\cdot j = 0$. For charge: $\partial_t\rho + \nabla\cdot J = 0$.

Vol. 2, 6.8 Maxwell's equations, conservation laws and heat

If events grow, $A_1 \subseteq A_2 \subseteq \cdots$, then $P(A_n) \to P(\bigcup A_n)$. If events shrink, $A_1 \supseteq A_2 \supseteq \cdots$, then $P(A_n) \to P(\bigcap A_n)$.

Vol. 2, 10.1 Probability as measure

$f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$: small changes of input near $a$ give small changes of output.

Vol. 2, 1.3 Limits and continuity with epsilon and delta

If $f$ is continuous on $[a, b]$, then $f$ is integrable on $[a, b]$.

Vol. 2, 1.5 The Riemann integral: area as a limit

If $f$ is continuous on a closed rectangle $Q$, then $f$ is integrable on $Q$.

Vol. 2, 6.3 Multiple integrals

If $K$ is compact and $f : K \to Y$ is continuous, then $f(K)$ is compact.

Vol. 2, 4.3 Compactness

 

Vol. 2, 4.5 Continuity between metric spaces

If $f : X \to Y$ is continuous and $W$ is open in $Y$, then $f^{-1}(W) = \{x : f(x) \in W\}$ is open in $X$.

Vol. 2, 4.4 Connectedness and path-connectedness

If $f$ is continuous at $a$ and $x_n \to a$, then $f(x_n) \to f(a)$.

Vol. 2, 4.3 Compactness

A continuous function on $[a, b]$ is uniformly continuous there.

Vol. 2, 1.3 Limits and continuity with epsilon and delta

A continuous map from a compact metric space to any metric space is uniformly continuous: one $\delta$ works for the whole set.

Vol. 2, 4.3 Compactness

If $f$ is continuously differentiable at $a$, then $f$ is differentiable at $a$, with $T_a(v) = \nabla f(a)\cdot v$.

Vol. 2, 5.3 The total derivative, the Jacobian matrix and tangent planes

A scalar field is continuously differentiable at $a$ if its partial derivatives exist in some ball around $a$ and are continuous at $a$.

Vol. 2, 5.3 The total derivative, the Jacobian matrix and tangent planes

A map $T : X \to X$ that brings every pair of points closer together by a fixed factor $q < 1$: $d(Tx, Ty) \le q\,d(x, y)$.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

The contrapositive of $P \Rightarrow Q$ is $\neg Q \Rightarrow \neg P$; it always has the same truth value as the original.

Vol. 2, 0.1 How to read a theorem

Let $P$ be stochastic and diagonalizable. If $1$ is an eigenvalue with a one-dimensional eigenspace and every other eigenvalue has size less than 1, then $P^kx_0$ approaches the same stationary distribution $\pi$ from every starting probability vector $x_0$.

Vol. 2, 2.11 Markov chains and PageRank

A sequence of vectors converges in $\mathbb R^n$ (for any of the 1-, 2- or max-distances) if and only if each coordinate converges as a sequence of real numbers.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

For continuous $f_n$ and $f$ on $[a, b]$: $d_\infty(f_n, f) \to 0$ if and only if $f_n \to f$ uniformly on $[a, b]$.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

If $\sum c_k x_0^k$ converges, then $\sum c_k x^k$ converges absolutely for every $|x| < |x_0|$.

Vol. 2, 1.6 Series, power series, and uniform convergence

 

Vol. 2, 1.2 Sequences and limits, from zero

 

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

If $a_n \to L$, some single number bounds every $|a_n|$.

Vol. 2, 1.2 Sequences and limits, from zero

If $x_n \to x$ in a metric space, then $(x_n)$ is Cauchy.

Vol. 2, 4.2 Convergence, completeness and the contraction mapping theorem

The converse of $P \Rightarrow Q$ is $Q \Rightarrow P$; it is a different statement and can be false when the original is true.

Vol. 2, 0.1 How to read a theorem

A function $f$ on a convex set with $f\big(tx + (1 - t)y\big) \le tf(x) + (1 - t)f(y)$ for all $x, y$ and $t \in [0, 1]$: the graph lies below every chord.

Vol. 2, 5.8 Constrained extrema and Lagrange multipliers

A set $S$ is convex if for any two points $p, q$ in $S$, the whole segment $(1 - t)p + tq$, $0 \le t \le 1$, lies in $S$.

Vol. 2, 6.2 Gradient fields and potentials

The unique numbers $(c_1, \dots, c_n)$ with $x = \sum c_i e_i$ for a chosen, ordered basis.

Vol. 2, 2.2 Span, independence, basis, dimension

$[x]_{\mathcal E} = C\,[x]_{\mathcal F}$, and $C$ is invertible with $C^{-1} = m_{\mathcal E \to \mathcal F}(I)$.

Vol. 2, 2.5 Matrices as maps, multiplication, change of basis

In an orthogonal basis, the coordinate of $x$ on $e_i$ is $(x, e_i)/(e_i, e_i)$; in an orthonormal basis it is $(x, e_i)$ itself.

Vol. 2, 2.9 Inner products, orthogonality, Gram–Schmidt, projection

For a subspace U and a vector v, the coset v + U is the set of all v + u with u in U: the subspace U moved by v.

Vol. 2, 3.7 Dual spaces and quotient spaces

Define v ∼ w when v − w ∈ U. This is an equivalence relation whose class of v is exactly the coset v + U. So v + U = w + U iff v − w ∈ U, and any two cosets are equal or disjoint.

Vol. 2, 3.7 Dual spaces and quotient spaces

A set is countable if its elements can be put in a list, first, second, third, with every element appearing exactly once; finite sets count too.

Vol. 2, 0.3 Sets, functions, and counting infinity

For an invertible $A$, the solution of $Ax = b$ has $x_j = \det A_j(b)/\det A$, where $A_j(b)$ is $A$ with column $j$ replaced by $b$.

Vol. 2, 2.7 Determinants

If $\det A \ne 0$ then the unique solution of $Ax = b$ is $x_j = \det A_j(b)/\det A$.

Vol. 2, 2.7 Determinants

A point $a$ where a differentiable scalar field has $\nabla f(a) = O$; Apostol calls it a stationary point.

Vol. 2, 5.6 Taylor's formula, the Hessian and extrema

For $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ in $\mathbb R^3$, $a \times b = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)$: a vector perpendicular to both, whose length is the area of the parallelogram with sides $a$ and $b$.

Vol. 2, 6.5 Surfaces, area and flux

For $F = (P, Q, R)$, $\operatorname{curl}F = \nabla\times F = (R_y - Q_z,\ P_z - R_x,\ Q_x - P_y)$: a vector field that records how much, and about which axis, $F$ spins at each point.

Vol. 2, 6.6 Curl, divergence, Stokes' theorem and Gauss's theorem

For a $C^1$ field $F$, a point $p$ and a unit vector $n$, the circulation of $F$ around the circle of radius $\varepsilon$ about $p$ in the plane perpendicular to $n$ (counterclockwise from the tip of $n$), divided by its area $\pi\varepsilon^2$, tends to $(\operatorname{curl}F)(p)\cdot n$ as $\varepsilon\to0$.