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Let $U=aT+bI$ on a nonzero finite-dimensional vector space over a field, with $a\ne 0$. An invariant block for $T$ is also invariant for $U$. If $v,Tv,\dots ,T^{d-1}v$ is its cyclic vector basis, then $v,Uv,\dots ,U^{d-1}v$ is another basis: the binomial theorem gives a triangular change-of-basis matrix with diagonal $1,a,\dots ,a^{d-1}$. The monic polynomial $g(t)=a^{d}f((t-b)/a)$ annihilates this block, and its degree is $d$, so it is the block's minimal polynomial. In the new cyclic basis the block is $C(g)$.

The substitution preserves divisibility of the monic invariant factors of a linear operator. Apply it to every block to obtain the rational canonical form of $U$. If $a=0$, then $U=bI$ is a scalar matrix instead: all invariant factors are $t-b$, one for each dimension. This exceptional case cannot be obtained by dividing by $a$.

Let $A$ act on a nonzero finite-dimensional vector space over a field $F$, with minimal polynomial an irreducible polynomial $f$ of polynomial degree $d$. The nonconstant invariant factors of a linear operator all divide $f$, so each equals $f$. Their degrees sum to the dimension of a vector space, giving $d\mid n$ and rational canonical form consisting of $n/d$ copies of the companion matrix $C_f$. Equivalently, make the vector space a module over the field $F[X]/(f)$ by letting $X$ act as $A$; a basis over this field gives the same block decomposition.

For $n\ge 2$, the rational homotopy group of a based topological space $X$ is the vector spaceHere ${\pi }_n(X)$ is an abelian group, so this tensor product is defined. For a finitely generated abelian group ${\pi }_n(X)\cong {\mathbb{Z}}^r\oplus T$ with $T$ finite, the result is ${\mathbb{Q}}^r$: finite-order classes disappear and the free generators acquire rational coefficients. More generally every torsion element maps to zero, since $ma=0$ implies $a\otimes q=(ma)\otimes (q/m)=0$ for $m\ne 0$.

For example, ${\pi }_3(S^2)\cong \mathbb{Z}$ gives ${\pi }_3(S^2)\otimes \mathbb{Q}\cong \mathbb{Q}$, whereas ${\pi }_4(S^3)\cong \mathbb{Z}/2$ becomes zero. The general tensor product definition also handles groups that are not finitely generated: for an Eilenberg–MacLane space $K(\mathbb{Q},n)$, its degree-$n$ rational homotopy group is still $\mathbb{Q}$.

For a simply connected CW complex, these vector spaces are the homotopy groups of its rationalization of a topological space. An arbitrary nonabelian fundamental group cannot be tensored with $\mathbb{Q}$ by this abelian group formula; the nilpotent-group localization used more generally in rational homotopy theory is a separate construction.

A rational-linear map is a linear map when its domain and codomain are regarded as vector spaces over the rational numbers. An additive map on a rational vector subspace of the real numbers is automatically rational-linear: integer homogeneity follows by repeated addition, and division by a positive integer gives rational homogeneity of an additive function.

In the simply connected homotopy category, a rational space is a simply connected CW complex whose positive homotopy groups are vector spaces over $\mathbb{Q}$. Multiplication by every nonzero integer on these groups is invertible. More generally one uses nilpotent spaces with localized fundamental group and higher homotopy groups; the simply connected version avoids local-coefficient twisting.

A real matrix is a matrix whose entries are real numbers. The vector space of $m\times n$ real matrices is denoted $M_{m,n}(\mathbb{R})$.

A real representation of a group $G$ is a homomorphism $G\to \mathrm{GL}(V)$ for a real vector space $V$.

If a real linear operator $A$ on an $n$-dimensional vector space has $n$ distinct real eigenvalues, any real square root $B$ commutes with $A$ and preserves each one-dimensional eigenspace. There $B$ is multiplication by a real $b_i$, hence ${\lambda }_i=b_i^2\ge 0$. Conversely, if all the eigenvalues are nonnegative, choosing $b_i=\sqrt{{\lambda }_i}$ on an eigenbasis supplies a real square root. Without simple spectrum the necessity fails: a real quarter-turn squares to $-I$.

A real vector space is a vector space over the field $\mathbb{R}$ of real numbers.

Consider the linear map $A:{\mathbb{R}}^3\to {\mathbb{R}}^2$ represented by the rectangular matrixTo compute its kernel, solve the homogeneous linear system $x+2y=0$, $y+z=0$. Taking $y=t$ gives $x=-2t$, $z=-t$, soThus $v$ represents a null direction of a linear map: decreasing the first input by two, increasing the second by one, and decreasing the third by one changes neither output. Solving the homogeneous linear system is the concrete computation behind a kernel; the column space instead decides which outputs are attainable.^[ref]

Now prescribe the nonzero output $b=(3,2{)}^T$. The linear system $Au=b$ becomes $x+2y=3$, $y+z=2$. Again take $y=t$:Direct substitution gives $A{u}_0=(3,2{)}^T$, and $A(u_0+tv)=A{u}_0+tAv=b$. Conversely, the two linear equations force exactly this form, so no solutions have been omitted. This fiber is an affine subspace of dimension one in the three-dimensional input vector space. It is parallel to the kernel, but does not pass through the origin. For instance, $u_0=(3,0,2{)}^T$ and $u_1=(1,1,1{)}^T$ both give $b$, while their difference $u_1-u_0=v$ gives zero. Adding $u_0$ and $u_1$ gives output $2b$, which is another way to see that this nonzero-output fiber is not a vector subspace.

The first and third columns of $A$ are $(1,0{)}^T$ and $(0,1{)}^T$, a basis of the output vector space. Hence $A$ has matrix rank two and reaches every output. The rank-nullity theorem predicts nullity $3-2=1$ before we solve any linear system. It provides a useful completeness check: one nonzero kernel vector already forms a basis here, so a second independent null direction of a linear map would signal a calculation error. In general, matrix rank $r$ for an $m\times n$ matrix leaves $n-r$ independent kernel directions, rather than $m-r$.^[ref]

Let $f,g:A\rightrightarrows B$ be a reflexive pair in an additive category with finite limits, with common section $r:B\to A$. Regard $f$ as source and $g$ as target. For arrows $x,y:C\to A$ satisfying $gx=fy$, defineThe identity at $b:C\to B$ is $rb$, and the inverse of $x$ is $rfx+rgx-x$. These operations define an internal groupoid. The finite limits hypothesis supplies the objects of composable arrows needed for the groupoid diagrams.

The source of the composite is $fx+fy-gx=fx$, and its target is $gx+gy-gx=gy$. The common-section identities prove both identity laws and both inverse laws. For a composable triple $x,y,z$, either parenthesization gives $x+y+z-rgx-rgy$, proving associativity. Bilinearity makes these formulas compatible with precomposition in $C$.

The limit hypothesis cannot simply be discarded. Take the full subcategory of finite-dimensional vector spaces whose dimensions are $3a+5b$, for nonnegative integers $a,b$. A projection $k^5\to k^3$ has a common section with itself, but its composable-pair pullback in a category in vector spaces has dimension $7$, which is not of the allowed form. Testing the pullback in a category property on $k^3$ would force that same dimension, so the smaller category has no such pullback in a category. The reflexive-pair groupoid formula in a preadditive category still makes sense on generalized arrows without asserting existence of this object.

A regularity structure is an algebraic framework for local expansions that can include singular distributions as well as smooth functions. Its formal data are $(A,T,G)$: a locally finite set $A\subset \mathbb{R}$ containing zero and bounded below; a graded vector space $T={⨁}_{\alpha \in A}{T}_{\alpha }$ of Banach spaces, with $T_0$ the constants; and a group $G$ of linear maps fixing $1$ such thatThus changing an expansion's base point modifies a symbol only through lower-homogeneity terms. Martin Hairer introduced this framework to treat classes of singular stochastic partial differential equations.^[ref]

Ordinary polynomials provide a complete elementary example. In one dimension take $A=\{0,1,2,\dots \}$, $T_n$ spanned by $X^n$, and realize $X^n$ around $x$ as $(z-x{)}^n$. The function $f(z)=z^2$ then has abstract local expansion $a_x=x^2+2xX+X^2$. To transfer the expansion from $y$ to $x$, substitute $X\mapsto X+x-y$. For example,The expression on the left is the expansion at $y=2$ transported to $x=1$. In particular, ${\Gamma }_{12}{X}^2-X^2=-2X+1$ has strictly lower degree, illustrating the defining triangular property.

A model of a regularity structure adds concrete distributions and analytic bounds to these abstract symbols. Writing their realizations as ${\Pi }_x$, compatibility requires ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$. For singular equations the symbols extend beyond monomials; constructing a model, proving reconstruction and controlling corrections are additional steps. The triple alone does not make arbitrary products of distributions meaningful.^[ref]

Let $U,V,X,Y$ take values in the vector space ${\mathbb{F}}_2^n$ over the two-element finite field. In this additive group, subtraction equals addition. Suppose $(U,V)$ is a C-relevant pair of random variables relative to $(X,Y)$, and take mutually independent copies $U_1,U_2,V_1,V_2$. Then $(U_1+U_2,V_1+V_2)$ is $2C$-relevant to $(X,Y)$.

For a proof, let $S=U_1+U_2$ and take an independent representative of $X$. The entropy submodularity for three independent sums givesso the definition of Entropic Ruzsa distance yieldsThe last step is the Entropic Ruzsa triangle inequality. Apply the same argument to $V_1+V_2,Y$ and add the inequalities.^[ref]

The binary finite field hypothesis makes the displayed self-sum argument agree with the difference convention in Entropic Ruzsa distance. On a general abelian group, signs must be retained explicitly.

A representation of a complex Banach algebra is a complex-linear multiplicative action on a vector space. For a unital action require $\pi (1)=I$; a nonunital algebra can be handled by extending the action to its unitization. A nonzero action is algebraically irreducible when it has no nonzero proper invariant vector subspace, equivalently when it defines a simple module.

An algebra representation is a unital algebra homomorphism into the endomorphisms of a vector space. It is equivalent to making that vector space a left module over the algebra, with $av=\rho (a)v$.

A representation of a quiver over a field assigns a vector space to each vertex and a linear map to each arrow. A morphism is a compatible family of vertexwise linear maps.

A representation over the rational numbers is a group representation on a finite-dimensional rational vector space. Choosing a basis gives rational matrices. It differs from a rational representation of an algebraic group, where rational refers to regular matrix-coefficient functions rather than the ground field. Scalar extension yields representations over every characteristic-zero field.

Representation theory describes abstract symmetries by linear maps on vector spaces. A group representation assigns an invertible linear map to each group element, respecting multiplication. The payoff is that a symmetry problem becomes a problem about matrices, invariant subspaces, and eigenvalues. Decomposing the action can reveal combinations of variables that a symmetric system cannot mix.

Take the cyclic group $C_3=\{e,g,g^2\}$, with $g^3=e$. On three real coordinates, let $g$ send $(a,b,c)$ to $(c,a,b)$:This permutation representation does more than relabel a list: it acts linearly on every possible assignment of values to the three positions. The vector $(1,1,1)$ stays fixed. The vector subspace $W=\{(a,b,c):a+b+c=0\}$ is also preserved, since a permutation does not change the sum.

Every vector separates uniquely into these two types:For example, $(4,1,1)=2(1,1,1)+(2,-1,-1)$. The first term belongs to a trivial representation; the second is a contrast with zero coordinate sum. This is a direct sum decomposition ${\mathbb{R}}^3=\mathbb{R}(1,1,1)\oplus W$. The symmetry preserves the mean and rotates the contrasts.

The rotation is literal after choosing the orthonormal basis $u=(2,-1,-1)/\sqrt{6}$, $v=(0,1,-1)/\sqrt{2}$ of $W$. In this basis,It is a rotation through $120$ degrees, and three applications return to the starting point. There is no nonzero invariant real line, because this rotation has no real eigenvector. Thus the two-dimensional action is an irreducible representation over the real numbers. Over the complex numbers it instead splits into two eigenlines with eigenvalues $e^{2\pi i/3}$ and $e^{-2\pi i/3}$. The coefficient field is part of an irreducibility claim.

Why would this decomposition matter for a physical calculation? Consider three equal masses with scalar displacements $x_i$ and the symmetric quadratic coupling energyThe equations of motion are $m\ddot{x}=-kLx$, whereis a graph Laplacian and $J$ is the all-ones matrix. The constant vector has eigenvalue zero: translating all coordinates equally costs no coupling energy. On $W$, $J$ vanishes and $L=3I$, so both independent contrast directions have angular frequency $\sqrt{3k/m}$. A three-coordinate coupled problem reduces to a free mean and two identical harmonic oscillators.

For initial displacement $x(0)=(4,1,1)$ and initial velocity zero, the exact solution is $x(t)=2(1,1,1)+\cos (\sqrt{3k/m}t)(2,-1,-1)$. The fixed mean and oscillating contrast can be read directly from the decomposition. More generally, a linear operator commuting with the group action carries its fixed-vector subspace into itself; with an invariant inner product, it preserves the corresponding orthogonal complement too.

For a finite group, averaging over the action constructs an invariant inner product over the real numbers or complex numbers. The resulting invariant complements explain Maschke's theorem: a finite-dimensional group representation decomposes into irreducible representations. Characters compress the action into traces and help identify those pieces. The assumptions matter; in modular representation theory the coefficient field can have characteristic dividing the group order, and such decompositions can fail. Chapter 4 of Pavel Etingof and coauthors' notes develops these mechanisms. ^[ref].

For a group representation of $G$ and a subgroup $H$, restriction keeps the vector space and lets only the elements of $H$ act. It is denoted ${\mathrm{Res}}_H^{G}V$.

A right inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfying $f\circ g={\mathrm{id}}_Y$. It forces $f$ to be a surjective function. A surjective linear map onto a finite-dimensional vector space has a linear right inverse: lift a basis of the target and extend by linearity.