Linear algebra

A Cartesian basis is an orthonormal basis associated with the perpendicular axes of a Cartesian coordinate system.

A bilinear form on a vector space $V$ over a field $F$ is a map $B:V\times V\to F$ that is linear in each argument separately.

A symmetric real bilinear form is positive-definite when $B(v,v)>0$ for every nonzero vector. It defines an inner product and the associated norm $\sqrt{B(v,v)}$. The complex Killing form is not positive-definite on the entire complex Cartan subalgebra; its restriction to the Euclidean subspace of a Cartan subalgebra is a real positive-definite form.

A bilinear form is alternating if $B(v,v)=0$ for every vector $v$. Expanding $B(v+w,v+w)$ implies $B(v,w)=-B(w,v)$. The converse requires characteristic different from two. In that characteristic a skew-symmetric matrix of odd integer size has zero determinant, since $\det A=\det (-A^T)=(-1{)}^n\det A$. Thus an alternating form on a finite odd-dimensional vector space is degenerate. A nondegenerate alternating form defines a symplectic vector space.

A scalar-valued map $b:V\times W\to F$ is bilinear when it is linear in either argument with the other held fixed. A bilinear form is the case $V=W$. Checking bilinearity is one of the algebraic steps in verifying a real inner product; being a symmetric bilinear form and having positive squared norm impose further conditions.

A bilinear form $B$ is symmetric when $B(v,w)=B(w,v)$ for every pair of vectors $v,w$.

A bilinear form $B:X\times X\to \mathbb{R}$ on a normed vector space is bounded when some $C<\infty$ satisfies $|B(u,v)|\le C\Vert u\Vert \Vert v\Vert$ for all $u,v\in X$.

A bilinear form on a normed vector space is continuous exactly when it is bounded. For the nontrivial direction, continuity at $(0,0)$ gives $r,s>0$ such that $|B(x,y)|\le 1$ whenever $\Vert x\Vert <r$ and $\Vert y\Vert <s$; scaling nonzero $u,v$ into these balls gives $|B(u,v)|\le 4(rs{)}^{-1}\Vert u\Vert \Vert v\Vert$.

A bilinear form $B$ on a normed vector space is coercive when some $\alpha >0$ satisfies $B(v,v)\ge \alpha \Vert v{\Vert }^2$ for every $v$.

A bilinear form is degenerate when its radical contains a nonzero vector.

The radical of a symmetric bilinear form $B$ isThe form is nondegenerate exactly when its radical is zero.

A bilinear form $B$ on a finite-dimensional vector space is nondegenerate whenequivalently when $w\mapsto B(-,w)$ is an isomorphism from $V$ to its dual space.

A symplectic vector space is a vector space equipped with a nondegenerate antisymmetric bilinear form. Every finite-dimensional symplectic vector space has even dimension: splitting off one symplectic two-plane leaves a smaller nondegenerate antisymmetric space, and induction completes the decomposition.

If $T$ preserves a nondegenerate bilinear form $\omega$ that is alternating and satisfies $T^2=I$ on a finite-dimensional vector space over a field of characteristic different from two, thenThe two eigenspaces are orthogonal for $\omega$, since $\omega (v_{+},v_{-})=\omega (T{v}_{+},T{v}_{-})=-\omega (v_{+},v_{-})$. Each restricted form is nondegenerate: a vector annihilating its own eigenspace also annihilates the other and hence all of $V$. Both dimensions are even, say $2a$ and $2b$. Consequently $\mathrm{tr}T=2a-2b=\dim V-4b$ as an integer obtained from the eigenspace dimensions. Over the real numbers this gives the congruence $\mathrm{tr}T\equiv \dim V(\mathrm{mod}4)$.

The symplectic orthogonal complement of a subspace $W$ is $W^{\omega }=\{v:\omega (v,w)=0\text{ for every }w\in W\}$. The nondegenerate bilinear form gives $\dim W^{\omega }=\dim V-\dim W$. An isotropic subspace of a symplectic vector space satisfies $W\subset W^{\omega }$; a Lagrangian subspace satisfies equality.

A subspace $W$ of a symplectic vector space $(V,\Omega )$ is isotropic when $\Omega {|}_{W\times W}=0$. It has at most half the dimension of $V$; equality makes it a Lagrangian subspace. This use of isotropic concerns an alternating bilinear form, rather than an isotropic quadratic form.

A symplectic basis $(e_1,\dots ,e_n,f_1,\dots ,f_n)$ has pairings $\omega (e_i,f_j)={\delta }_{ij}$ and all pairings between two $e_i$ or two $f_i$ equal to zero.

If $B_1$ is a nondegenerate bilinear form on a finite-dimensional vector space and $B_2$ is any bilinear form, there is a unique linear map $\alpha$ such thatThe right radical of $B_2$ is $\ker \alpha$.

The inertia of a real symmetric bilinear form is the triple giving the numbers of positive, negative, and zero squares in a diagonalization. Sylvester's law of inertia makes it independent of the chosen basis.

For a real symmetric bilinear form, let $n_{+}$ and $n_{-}$ be the dimensions of its positive and negative diagonal subspaces. This pair gives the first two components of the inertia of a bilinear form and is independent of basis by the Sylvester's law of inertia. For a nondegenerate form their sum is the full dimension; otherwise the nullity is an additional invariant. Another convention calls the difference $n_{+}-n_{-}$ the signature of a quadratic form. The pair convention describes the positive and negative harmonic spaces of the real de Rham intersection form in dimension four.

A complex sesquilinear form is linear in one argument and conjugate-linear in the other. Here the first argument is taken to be linear.

A Hermitian form is a complex sesquilinear form $H$ satisfying $H(w,v)=\overline{H(v,w)}$.

The radical of a Hermitian form is the vector subspace orthogonal to the entire space. The form descends to the quotient vector space by its radical: adding a radical vector to either argument does not change the value. For a positive semidefinite Hermitian form, the Cauchy-Schwarz inequality identifies this radical with its zero-norm vectors, so the quotient form is positive definite. This conclusion does not hold for a general indefinite Hermitian form: a zero-norm vector need not belong to its radical.

A Hermitian form is positive semidefinite when $h(v,v)\ge 0$ for every vector. It need not define an inner product, because a nonzero vector may have zero squared norm. The Cauchy-Schwarz inequality still holds: applying nonnegativity to $v+zw$ and minimizing the quadratic expression in $z$ gives $|h(v,w){|}^2\le h(v,v)h(w,w)$ when $h(w,w)>0$; when $h(w,w)=0$, varying $z$ forces $h(v,w)=0$. Thus its zero-norm vectors are exactly the radical of a Hermitian form. Quotienting this radical of a Hermitian form gives a positive inner product; taking its Hilbert space completion then gives a Hilbert space. This is the final positivity step in the Gupta-Bleuler null-state quotient.

An indefinite Hermitian form takes both positive and negative values on $⟨v,v⟩$ for nonzero vectors. For example, $-|v_0{|}^2+|v_1{|}^2+|v_2{|}^2+|v_3{|}^2$ is indefinite. It is not a positive inner product defining a Hilbert space, and a nonzero vector can have zero norm without being orthogonal to all vectors. A positive semidefinite restricted form becomes a positive inner product only after quotienting its radical of a Hermitian form.

For a basis $(v_i)$, the matrix $A$ of a Hermitian form has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^{T}A\overline{y}$.

A vector space consists of vectors that may be added and multiplied by scalars, subject to the vector-space axioms.

A vector space with scalar field $\mathbb{Q}$ is equivalently a torsion-free divisible Abelian group. An embedding of such groups automatically respects rational scalar multiplication because solutions to $nb=ma$ are unique.

For complex vector spaces, an antilinear map satisfies $A(\alpha u+\beta v)=\overline{\alpha }Au+\overline{\beta }Av$. It is linear over the real numbers and conjugates complex scalar multiplication. This differs from two conjugate linear operators, which are related by similarity rather than conjugate-linearity.

An antiunitary operator between complex Hilbert spaces is a surjective antilinear map satisfying the displayed inner product identity. It preserves norms but conjugates scalars. In a chosen orthonormal basis it is $UK$, with $U$ a unitary operator and $K$ coefficientwise complex conjugation. Conversely any such $UK$ is antiunitary. Antilinearity by itself does not preserve norms.

A vector space over the field of complex numbers. A complex-linear map preserves addition and multiplication by complex scalars.

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

For real vectors of equal length and equal sum, write $x\prec y$ when ${\sum }_{j=1}^k{x}_j^{\downarrow }\le {\sum }_{j=1}^k{y}_j^{\downarrow }$ for every proper initial segment; the downward arrow means decreasing rearrangement. Majorization compares how unevenly a fixed total is distributed. The uniform probability vector is majorized by every probability vector, while $(1,0,\dots ,0)$ majorizes every probability vector. The order controls exact pure-state entanglement conversion in Nielsen's pure-state conversion theorem.

If ${\mathbb{F}}_q$ is a finite field, a vector space of finite dimension $d$ over ${\mathbb{F}}_q$ has $q^d$ elements. Every infinite-dimensional vector space over ${\mathbb{F}}_p$ is an infinite elementary abelian $p$-group under addition.

A subset of ${\mathbb{F}}_q^n$ is a finite-field Kakeya set when it contains an affine line in a vector space in every one-dimensional direction. The line position may depend on the direction. The polynomial method in combinatorics yields the finite-field Kakeya polynomial bound on its cardinality.

For a finite-field Kakeya set $A\subseteq {\mathbb{F}}_q^n$, a smaller cardinality would give a nonzero polynomial of total degree of a polynomial at most $q-1$ vanishing on $A$, by the dimension of a bounded-total-degree polynomial space. Its restriction to each complete affine line in a vector space has $q$ roots of a polynomial and degree less than $q$, hence vanishes identically. The top homogeneous polynomial part then vanishes in every direction. A polynomial of degree less than $q$ in each variable cannot vanish everywhere on ${\mathbb{F}}_q^n$ unless it is zero, giving a contradiction. For the prime field, this is the polynomial nonvanishing below the field size.

The linear span of a subset $S$ of a vector space is the set of all finite linear combinations of elements of $S$.

A subset $S$ of a vector space is a spanning set when its linear span is the whole vector space.

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 vector subspace is a subset closed under vector addition and scalar multiplication, and is therefore itself a vector space.

A flag is a strictly nested chain of vector subspaces of a fixed vector space. A complete flag contains a subspace of every possible dimension. For an action by linear maps, an invariant flag means that each of its vector subspaces is an invariant subspace. A basis adapted to an invariant complete flag makes all these maps upper triangular, connecting complete flags to simultaneous triangularization of a Lie algebra representation.

A complete flag in an $n$-dimensional vector space is a nested sequence of vector subspaces with $\dim V_i=i$. An action preserves such a flag exactly when its matrices are upper triangular in a basis adapted to the flag. The Lie theorem supplies an invariant complete flag for every complex finite-dimensional Lie algebra representation of a solvable Lie algebra.

The fixed coordinate subspace of dimension $k$ in ${\mathbb{R}}^p$ consists of vectors whose coordinates after the first $k$ vanish. Restricting a quadratic form to it uses only the leading $k$ by $k$ matrix block. It specifies one fixed support; allowing every support of size $k$ would introduce another combinatorial counting step.

The intersection of any family of vector subspaces of a common vector space is the set of vectors belonging to every member of the family. It is again a vector subspace.

The sum of vector subspaces $U,W\le V$ isIt is the smallest vector subspace containing both $U$ and $W$.

A hyperplane is a vector subspace of codimension one. A projective hyperplane in ${\mathbb{P}}^n$ is the zero set of one nonzero homogeneous linear form.

A closed vector subspace of a topological vector space is both a vector subspace and a closed set. The kernel of a continuous linear map is a closed vector subspace.

If $E$ is a nonzero closed linear subspace of a normed ambient space and $F$ is finite-dimensional with $E\cap F=\{0\}$, the unit sphere of $E$ has positive distance from $F$. Otherwise a sequence of nearly coincident points yields a bounded sequence in $F$, hence a convergent subsequence; closedness of $E$ puts its norm-one limit in both vector subspaces. In applications to a bidual space, completeness of the Banach space makes its canonical image closed.

An affine subspace of a vector space is a translate $v+W$ of a vector subspace $W$. Its dimension is the dimension of $W$.

In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.

A linear combination of vectors $v_1,\dots ,v_n$ is a sum $a_1{v}_1+\cdots +a_n{v}_n$ with scalar coefficients $a_i$.

A scalar multiplying a specified term in a linear combination, polynomial or power series. When the term family has linear independence, the coefficients are unique; uniqueness of finite Laurent polynomial coefficients follows after multiplying by a sufficiently large power to obtain an ordinary polynomial.

Scalar multiplication is the vector space operation $(a,v)\mapsto av$ taking a scalar and a vector to a vector.

A scalar multiple of a vector $v$ is a vector of the form $av$ for a scalar $a$.

The dimension of a vector space is the cardinality of any of its bases.

A vector space is finite-dimensional when it has a finite set as a basis.

An infinite-dimensional vector space has no finite basis, so its dimension is an infinite cardinal.