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A projective representation is a group homomorphism into the projective linear group of a vector space. Choosing invertible linear representatives makes multiplication hold up to nonzero scalar factors. A projective unitary representation chooses unitary operators as representatives, whose scalar factors have modulus one.

For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]

A proper vector subspace of a vector space is one strictly smaller than the ambient space. In a finite-dimensional vector space, this is equivalent to having smaller dimension: a basis of the subspace can be extended to a basis of the ambient space, and strict inclusion requires at least one additional vector.

For even $d=2n$ with $n\ge 2$, the roots of $X^{q^d}+X^{q^2}+TX$ form an $n$-dimensional vector space over ${\mathbb{F}}_{q^2}$. Ratios $(c\alpha )/\alpha$ for a nonzero root $\alpha$ show that the splitting field contains every $c\in {\mathbb{F}}_{q^2}$. The subgroup fixing that quadratic constant extension is normal of index two and lies in $GL(n,q^2)$; applying projective transitivity over ${\mathbb{F}}_{q^2}(T)$ makes it contain $SL(n,q^2)$. It therefore normalizes that special linear group, while the full group acts semilinearly.

A quadratic form on a vector space over a field $F$ is a function $Q:V\to F$ satisfying $Q(av)=a^{2}Q(v)$ whose polar bilinear form of a quadratic form $B_Q(v,w)=Q(v+w)-Q(v)-Q(w)$ is a bilinear form. In finite dimension, it is represented in coordinates by a homogeneous degree-two polynomial, or by $Q(v)=v^{T}Av$ with a suitable matrix $A$.

When the characteristic is not two, one may choose $A$ to be a symmetric matrix and recover it from $B_Q/2$. In characteristic two, the symmetric-matrix restriction would lose examples: $Q(x,y)=xy$ has an upper-triangular representing matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, while $v^{T}Av$ for a symmetric matrix has no $xy$ cross term. Its polar form is $B_Q((x,y),(u,v))=xv+yu$, an alternating bilinear form. The distinction matters when studying quadratic forms over finite fields.^[ref]

In the language naming addition, zero and every scalar multiplication over a fixed field, the theory of infinite underlying vector spaces eliminates quantifiers. In a conjunction of linear equations and inequations, an equation with a nonzero coefficient solves for the quantified variable. Otherwise the variable is excluded from only finitely many values, so infinitude supplies a witness. Disjunction and induction finish elimination. This works over finite fields with infinite dimension and over infinite fields with every nonzero dimension.

In a torsion-free divisible Abelian group, division by a nonzero integer is unique. The rational scalar operations are therefore a definitional expansion. Formulas reduce to Boolean combinations of rational linear equations: eliminate an existential variable in a conjunction by solving any equation with nonzero coefficient; if none fixes the variable, the finitely many nontrivial disequalities exclude only finitely many choices in an infinite group. Equations with zero coefficient reduce to parameter conditions.

Thus over a finite parameter set a complete one-variable type either specifies an element of its linear span, or excludes every element of that span. An uncountable rational vector space realizes the latter type because a finite-parameter span is countable. It is consequently an omega-saturated model in the additive language.

For $q\ne 0$, define a vector space with basis $u^i{v}^j$, $i\in \mathbb{Z}$, $j\ge 0$, and product $(u^i{v}^j)(u^k{v}^{\ell })=q^{-jk}{u}^{i+k}{v}^{j+\ell }$. Bilinearity of the exponent verifies associativity. This constructs the algebra and proves the basis independently of its presentation. It localizes the quantum plane at powers of $u$ and gives a quotient witness for noncommutativity and noncocommutativity of the coordinate Hopf algebra of quantum SL2.

A quasi-norm on a vector space is nonnegative, vanishes only at zero, is absolutely homogeneous, and satisfies a triangle inequality with one constant factor $C\ge 1$. For $0<p<1$, $(\int |f{|}^p{)}^{1/p}$ on an Lp space is a quasi-norm: $\Vert f+g{\Vert }_p^p\le \Vert f{\Vert }_p^p+\Vert g{\Vert }_p^p$ gives a quasi-triangle inequality. Measure-preserving composition preserves this quantity even though the Minkowski inequality is unavailable below exponent one.

Its path algebra is $k[t]$. Finite-dimensional representations are vector spaces with a single endomorphism. Thus oriented cycles can give an infinite-dimensional algebra even when the representations studied are finite-dimensional.

For a Lie algebra ideal $I$, the quotient vector space $\mathfrak{g}/I$ has Lie bracket $[x+I,y+I]=[x,y]+I$. The ideal condition makes this independent of representatives. Its Lie algebra ideals correspond to the ideals of $\mathfrak{g}$ containing $I$, by inverse image under the quotient Lie algebra homomorphism.

For a vector subspace $W\le V$, the quotient vector space $V/W$ consists of cosets $v+W$, with vector operations induced from $V$.

A subset $C$ of a real vector space is radially open when, for each $x\in C$ and each vector $v$, the points $x+tv$ remain in $C$ for all sufficiently small real $t$. No ambient topology is required. If $C$ is also convex and contains zero, it is absorbing and its Minkowski functional $p_C(x)=\inf \{s>0:x\in sC\}$ is finite and sublinear, with $C=\{p_C<1\}$. Convexity gives subadditivity; radial openness permits a slight dilation of any point of $C$, giving the strict inequality.

Take independent uniform samples from a $d$-dimensional vector space over the two-element field. At current matrix rank $r$, the existing span contains $2^r$ of the $2^d$ possible vectors, so a new sample increases the matrix rank with probability $1-2^{r-d}$. The mean waiting time for that increase is $1/(1-2^{r-d})$ by the geometric distribution. Adding the waiting times gives $\mathbb{E}M=d+{\sum }_{j=1}^d(2^j-1{)}^{-1}<d+2$. For $d\ge 1$, the bound follows by separating $j=1$ and using $(2^j-1{)}^{-1}<2^{1-j}$ for $j\ge 2$; for $d=0$, no samples are needed.

After $m=d+t$ samples, failure to span implies that some nonzero linear functional vanishes on all samples. Each fixed nonzero functional does so with probability $2^{-m}$; the union bound over the $2^d-1$ possibilities gives failure less than $2^{-t}$. This provides both an expected sample count and a fixed-budget high-success version of Simon's algorithm.

Let $U,V$ be nonzero finite-dimensional complex vector spaces, and let $\varphi :U\otimes V\to W$ have injective restrictions when either nonzero factor is fixed. It induces a map from ${\mathbb{CP}}^{a-1}\times {\mathbb{CP}}^{b-1}$ to ${\mathbb{CP}}^{r-1}$, where $a=\dim U$, $b=\dim V$, and $r=\mathrm{rank}\varphi$. Pullback of the degree-two generator is $x+y$, since both factor restrictions are projective linear embeddings. Its $(a+b-2)$th power is $\left(\genfrac{}{}{0ex}{}{a+b-2}{a-1}\right)x^{a-1}{y}^{b-1}\ne 0$ in integral cohomology. The target must therefore have complex dimension at least $a+b-2$. Positivity of both factor dimensions is necessary; a zero factor can make the claimed inequality false.

For linear maps $B:U\to V$ and $A:V\to W$ with finite-dimensional $V$, the rank inequality for a composition isRestrict $A$ to $\mathrm{im}B$. The rank-nullity theorem gives $\mathrm{rank}(AB)=\dim \mathrm{im}B-\dim (\mathrm{im}B\cap \ker A)$, and the intersection has dimension at most $\dim \ker A$.

For example, the product of two rank-four endomorphisms of a five-dimensional vector space has rank at least three. The bound controls how much rank can be lost when the image of one map meets the kernel of the next.

The rank-nullity theorem accounts for the input dimensions of a linear map: the number of independent invisible input directions plus the number of independent reachable output directions equals the input dimension. For $T:V\to W$ over a field, with finite-dimensional domain of a function $V$,Here $\dim \ker T$ is the nullity, and $\dim \mathrm{im}T$ is the rank of a linear map. The kernel lives in the input vector space, while the image of a linear map lives in the output vector space. For an $m$-by-$n$ matrix of matrix rank $r$, the formula is $n=(n-r)+r$: the number of columns counts input coordinates. ^[ref]

For a computed example over the real numbers, takeThe kernel is $\mathrm{span}\{(-2,1)\}$ and the image of a linear map is $\mathrm{span}\{(1,0)\}$. Each has dimension one, so the two input dimensions split as $2=1+1$. The vector $(3,1)$ can be writtenThe second summand is undetectable: $T(3,1)=T(1,2)$. The picture uses the standard dot product to choose a visible direction perpendicular to the kernel; this is a convenient geometric choice, not an extra hypothesis of the rank-nullity theorem.

This explains a practical tradeoff in a linear system. If $T(x_0)=b$, every other input with output $b$ is $x_0+k$ for $k\in \ker T$. The nullity therefore counts independent freedom in an existing solution; the rank counts independent reachable output directions. In the example, $T(x,y)=(3,0)$ has all solutions $(x,y)=(3-2t,t)$, while $T(x,y)=(3,1)$ has none. The cokernel records this failure of reachability.

For an $m$-by-$n$ matrix, injectivity requires matrix rank $n$, and surjectivity requires matrix rank $m$. When $m\ne n$, these conditions differ. For example, $B:{\mathbb{R}}^3\to {\mathbb{R}}^2$ with $B(x,y,z)=(x+2y,0)$ has rank one and nullity two, because $\ker B=\mathrm{span}\{(-2,1,0),(0,0,1)\}$. The input accounting is $3=2+1$, although only one of two output directions is reached.

Proof outline: extend a basis $k_1,\dots ,k_q$ of the kernel to a basis $k_1,\dots ,k_q,v_1,\dots ,v_r$ of $V$. Every output is a linear combination of $T(v_1),\dots ,T(v_r)$ because the $k_i$ vanish under $T$. If a linear combination of the $T(v_j)$ is zero, the corresponding combination of the $v_j$ lies in the kernel; linear independence of the full basis forces all its coefficients to vanish. Thus the added basis vectors give a basis of the image of a linear map. Counting them proves the formula.

Over a field of characteristic different from two, the rank of a quadratic form is the rank of a linear map associated with its symmetric bilinear form, equivalently the matrix rank of a symmetric coefficient matrix. For an $n$-dimensional vector space this is $n-\dim \mathrm{rad}B$. A change of basis acts by matrix congruence, which preserves matrix rank.

For example, $q(x,y,z)=x^2-y^2$ has rank two, although it vanishes at the nonzero vector $(1,1,0)$. Its radical of a bilinear form is the $z$-axis; an isotropic vector need not lie in that radical. In characteristic two, the polar bilinear form of a quadratic form can lose square terms: $q(x)=x^2$ is nonzero but its polar form is zero. The rank of that polar form alone therefore does not record all quadratic information; conventions for quadratic rank in characteristic two require separate care.

The rank of a vector bundle at a point is the dimension of its fiber as a vector space. A vector bundle trivialization identifies nearby fibers with the same vector space, so this rank is locally constant. A rank-$r$ bundle has fibers of dimension $r$ everywhere.

The rational canonical form of a linear operator on a finite-dimensional vector space over a field is a direct sum $C(f_1)\oplus \cdots \oplus C(f_r)$ of companion matrices, where the monic nonconstant invariant factors of a linear operator satisfy $f_1\mid \cdots \mid f_r$. Their degrees sum to the dimension of the space. It exists over any field and classifies similar matrices, without requiring their eigenvalues to lie in that field. It comes from the structure theorem for finitely generated modules over a principal ideal domain, applied to the module in which the indeterminate acts as the given operator.