Vector space ($V$)

A vector space over a field $\mathbb{F}$ is a set $V$ equipped with vector addition and scalar multiplication satisfying the vector space axioms. Its vectors can represent directed displacements, lists of measurements, polynomials, or entire functions. The common structure is that adding two elements and multiplying an element by a scalar remain meaningful operations inside the same space.

For example, ${\mathbb{R}}^2$ uses coordinatewise addition and scaling. Withwe have $v=2b_1+b_2$. Moving twice along $b_1$ and once along $b_2$ reaches the same vector as five horizontal and four vertical steps. The two descriptions refer to different bases of the same space. A basis makes such descriptions unique; a spanning set makes them possible.

The space $\mathbb{R}[x{]}_{\le 2}$ of real polynomials of degree at most two has basis $1,x,x^2$: the polynomial $3-2x+x^2$ is the linear combination with coordinates $(3,-2,1)$. The zero vector here is the zero polynomial, and addition combines coefficients. A derivative is a linear map on this space; its matrix records how it acts on the three basis vectors. The same coordinate machinery therefore handles both geometric motion and operations on functions.

The choice of field matters. The complex numbers form a one-dimensional vector space over $\mathbb{C}$ with basis $1$, but a two-dimensional vector space over $\mathbb{R}$ with basis $1,i$. The question "how many independent directions?" has an answer only after the allowed scalars are specified. Likewise, polynomials of degree exactly two do not form a vector space: subtraction can remove the leading term, and the zero polynomial is absent.

The usefulness of the abstraction is that a linear map is determined by its values on a basis. Once those images are known, linear combinations give its action on every vector. The rank-nullity theorem relates the number of attainable outputs to the number of invisible input directions, while change of basis lets us choose coordinates suited to the problem. Eigenvectors, for example, choose directions on which repeated application of a linear operator becomes scalar multiplication.

A historical source is Hermann Grassmann's Die lineale Ausdehnungslehre (1844). Its full title advertises applications to other branches of mathematics, statics, mechanics, magnetism and crystallography; the original edition is digitized by ETH-Bibliothek. This gives an early setting for treating geometric quantities algebraically across different applications, rather than attributing the modern axiom list to that title alone.^[ref]

For a nonzero vector space with an infinite field or infinite basis, cardinality is the displayed maximum, since vectors are finite linear combinations of basis vectors. Over a finite field, infinite underlying size forces infinite dimension. Over a countably infinite field, every positive finite dimension is already countably infinite. The theory of infinite underlying spaces is therefore categorical in every infinite cardinal for a finite field, but only in uncountable cardinals for a countably infinite field.

A module over a division ring has the usual basis and linear independence theory of a vector space, with the side of scalar multiplication specified. For a right module, a finite relation has the form ${\sum }_i{v}_i{a}_i=0$. Every nonzero endomorphism of a one-dimensional right module is invertible. If $V$ is a simple module on the left over $R$, evaluation makes it a right vector space over the opposite ring ${\mathrm{End}}_R(V{)}^{\mathrm{op}}$, a division ring by Schur's lemma. This is the scalar structure used by the Jacobson density theorem.

The complexification of a real vector space $V$ allows complex scalar coefficients while preserving its real linear relations. It can be constructed as the tensor product $V{\otimes }_{\mathbb{R}}\mathbb{C}$, or as formal sums $u+iv$ with $u,v\in V$. A real basis of $V$ becomes a complex basis of $V_{\mathbb{C}}$, so its complex dimension equals the original real dimension. Every real linear map extends uniquely by complex linearity. In particular a real representation extends by viewing its real matrices as complex matrices. Complexifying the realification of a complex representation produces the original complex representation together with its complex conjugate; the coordinate change $(u,v)\mapsto (u+iv,u-iv)$ exhibits the two blocks.

Addition makes $V$ an Abelian group: for all $u,v,w\in V$, $(u+v)+w=u+(v+w)$ and $u+v=v+u$, there is a zero vector $0$, and every $v$ has an additive inverse $-v$. Scalar multiplication satisfies, for $a,b\in \mathbb{F}$,Both operations take values in $V$. These requirements make manipulating linear combinations independent of how sums are grouped or expanded.

A sequence space is a vector space of scalar sequences, closed under coordinatewise addition and scalar multiplication. Examples include the l-p sequence space, the l-infinity sequence space and spaces of convergent sequences.

On ${\ell }^1$, $q$ is a norm, bounded above by $\Vert x{\Vert }_1$. The partial-sum map $Jx=({\sum }_{n\le k}{x}_n{)}_k$ is an isometric linear embedding into the space $c$ of convergent sequences with the supremum norm. Its range contains every eventually constant sequence: take successive differences of such a sequence to obtain a finitely supported element of ${\ell }^1$. These sequences are dense in $c$, so the completion of a normed space is $c$.

The range is proper: partial sums of the alternating harmonic series converge but their successive differences do not belong to ${\ell }^1$. The finite truncations of these differences are $q$-Cauchy and cannot have a limit in ${\ell }^1$. Thus this normed version of ${\ell }^1$ is incomplete, despite the completeness of its usual norm.

The zero vector of a vector space is its additive identity: $v+0=0+v=v$ for every vector $v$. It is unique, because if $0^{\prime }$ is another such identity then $0=0+0^{\prime }=0^{\prime }$. Distributivity gives $\alpha 0=\alpha (0+0)=\alpha 0+\alpha 0$, so $\alpha 0=0$ after cancellation. In a coordinate vector space it has every coordinate zero.

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 endomorphism $J$ that is an antilinear map with $J^2=-I$ equips a complex vector space with a quaternionic structure: complex scalar multiplication supplies the quaternion generator $i$ and $J$ supplies $j$, satisfying $Ji=-iJ$. In finite dimensions the complex dimension must be even. Indeed, there is no $J$-invariant complex line: $Jv=cv$ for $v\ne 0$ would imply $-v=|c{|}^{2}v$. Consequently $v,Jv$ span a $J$-invariant complex two-plane; passing to its quotient vector space and repeating proves evenness.

In complex dimension four the invariant complex two-planes are quaternionic lines, forming the quaternionic projective space ${\mathbb{HP}}^1\cong S^4$. This is the Euclidean reality structure on twistor space used in twistor theory.

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.

For decreasing positive sequences in ${\ell }^1$, write $x\prec y$ if every initial partial sum of $x$ is at most the corresponding partial sum of $y$ and the finite total sums agree. The finiteness requirement matters: equality of two divergent totals interpreted as infinity is too weak to guarantee a doubly stochastic realization of summable-sequence majorization.

Decreasing strictly positive summable sequences with equal totals and $x\prec y$ admit an infinite doubly stochastic matrix realizing the displayed relation. Choose $k$ with $y_k\ge x_1>y_{k+1}$ and average the pair to obtain $x_1$ and $y_k+y_{k+1}-x_1$. The remaining decreasing sequence majorizes the remaining target sequence: short initial sums contain values at least $x_1$, and longer sums are the old sums minus $x_1$. Repeat on the tail, freezing one normalized finite-support row at each stage. Limit column sums are at most one; Tonelli's theorem, equal finite totals and strict positivity of every $y_j$ force every column sum to equal one.

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.

For infinite spaces over a field of size $q$, two-variable types are determined by the kernel of the linear map sending $(a,b)$ to $ax+by$. Its ranks give the zero pair, $q+1$ one-dimensional directions and the independent-pair type. All $q+3$ types are isolated by finite formulas. Quantifier elimination makes the Lindenbaum algebra their full power-set Boolean algebra. The one-variable algebra has just the zero and nonzero atoms, hence four Elements.

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.

Work in the affine plane over a finite field ${\mathbb{F}}_q$ of odd characteristic. The union of the affine lines $y=mx-m^2/4$, for $m\in {\mathbb{F}}_q$, contains a line of every finite slope. Completing the square givesFor a fixed $x$, the union therefore contains exactly the $(q+1)/2$ values of $y$ for which $x^2-y$ is a square, including zero. Adding the vertical affine line $x=0$ completes all directions. The resulting finite-field Kakeya set has cardinalityIts density tends to one half as $q$ grows through odd prime powers. Geometrically the finite-slope lines are the tangents to the parabola $y=x^2$, because the tangent at $x=m/2$ has the displayed equation.

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 $S$. It is the smallest vector subspace containing $S$: every vector subspace containing the original vectors must contain their sums and scalar multiples.

In ${\mathbb{R}}^3$, the linear span of $(1,0,0)$ is a line, the linear span of $(1,0,0),(0,1,0)$ is the plane $z=0$, and adding $(0,0,1)$ gives all of ${\mathbb{R}}^3$. Adding $(2,0,0)$ to the first set changes none of its attainable vectors, because it is already a scalar multiple of $(1,0,0)$. Thus a spanning set can contain redundant generators; linear independence identifies when this redundancy is absent. In an infinite-dimensional space, an algebraic linear span still uses finite sums; a closed linear span can also include limits in a specified norm.

In a normed vector space, the closed linear span of $S$ is the closure of its linear span in the norm topology. It includes limits of finite linear combinations. In a Hilbert space, an infinite orthonormal basis spans in this closed sense, allowing norm-convergent series in addition to finite sums.

Let $M$ be the closed span of these vectors in the real l2 sequence space. A vector $\beta$ is orthogonal to $M$ exactly when ${\beta }_{k+2}-2r{\beta }_{k+1}+{\beta }_k=0$. The characteristic roots are $r\pm \sqrt{r^2-1}$; one grows geometrically and the other, $\rho$, lies in $(0,1)$. Square summability eliminates the growing solution, so $M^{\perp }=\mathrm{span}\{({\rho }^{j-1}{)}_{j\ge 1}\}$. Hence $M$ is a closed hyperplane, namely the kernel of the bounded functional $\alpha \mapsto {\sum }_j{\alpha }_j{\rho }^{j-1}$. This combines a recurrence with density by annihilators.

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 $V$.

The quotient map of vector spaces $\pi :V\to V/U$ sends a vector $v$ to its coset $v+U$, where $U$ is a linear subspace. It is a surjective linear map with kernel $U$: two vectors have the same image exactly when their difference lies in $U$.

For $V={\mathbb{R}}^2$ and $U=\mathrm{span}(1,0)$, the quotient retains only the second coordinate, so every horizontal line represents one quotient vector. More generally, a linear map $f:V\to W$ vanishing on $U$ factors uniquely as $f=\overline{f}\circ \pi$. The formula $\overline{f}(v+U)=f(v)$ is well defined because changing the representative by an element of $U$ leaves its value unchanged.

For a linear map $T:V\to V$ and an invariant subspace $U$, the rule $\overline{T}(v+U)=T(v)+U$ defines a linear map on the quotient vector space $V/U$. Indeed, two representatives differ by an element of $U$, so their images also differ by an element of $U$. Conversely, this rule is well-defined only if $T(U)\subseteq U$, by comparing every $u\in U$ with zero.

For a vector subspace $W$ of $X$, its codimension is the dimension of the quotient vector space $X/W$. This definition also applies when $X$ is infinite-dimensional. If $X=W\oplus E$ is a direct sum, the map $e\mapsto e+W$ is a linear isomorphism from $E$ to $X/W$, so ${\mathrm{codim}}_{X}W=\dim E$. In particular, the kernel of a nonzero linear functional has codimension one: normalize $Te=1$ and write $x=(x-(Tx)e)+(Tx)e$.

For fixed $u\ne 0$, the parallel affine lines $l_x=x+\mathrm{span}(u)$ are precisely the cosets of the indicated quotient vector space. Changing representatives adds a multiple of $u$, so $l_x+l_y=l_{x+y}$ and $a{l}_x=l_{ax}$ are well-defined. The zero is the whole line $l_0$, not a single point. If $(u,b_1,b_2)$ is a basis, the two line classes $l_{b_1},l_{b_2}$ form a basis of the quotient by spanning and independence modulo $\mathrm{span}(u)$.

A vector subspace is a nonempty subset of a vector space closed under vector addition and scalar multiplication, and is therefore itself a vector space. Equivalently, whenever $u,v$ belong to the subset, so does $au+bv$ for all allowed scalars $a,b$. In particular it contains the zero vector.

For example, the plane $W=\{(x,y,z):x+y+z=0\}\subseteq {\mathbb{R}}^3$ is a vector subspace, with basis $(1,-1,0),(1,0,-1)$. Indeed every element is $(-y)(1,-1,0)+(-z)(1,0,-1)$. The parallel plane $x+y+z=1$ misses the zero vector and is an affine subspace instead. Homogeneous constraints produce vector subspaces: $W$ is the kernel of the linear map $(x,y,z)\mapsto x+y+z$. This lets us describe all solutions of a homogeneous linear equation through a basis of its kernel.

For a chosen basis $(e_1,\dots ,e_n)$, the vector subspace spanned by a specified collection of basis vectors. In the standard basis of Euclidean space, its vectors have zero coordinates outside $\mathcal{B}$. The block coordinate gradient is the orthogonal projection of the full gradient onto this subspace. The fixed coordinate subspace consisting of the first $k$ coordinates is a particular example.

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.

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 $W$.

For vector subspaces $U\subseteq W\subseteq V$ of a finite-dimensional vector space, such a $W^{\prime }$ always exists. Extend a basis of $U$ to one of $W$, then extend again to one of $V$. Let $W^{\prime }$ be spanned by the original basis of $U$ together with the vectors added in the second extension. Uniqueness of coordinates gives the displayed intersection and sum. Equivalently, choose a direct sum complement $C$ with $V=W\oplus C$ and put $W^{\prime }=U\oplus C$. Necessarily $\dim W^{\prime }=\dim V-\dim W+\dim U$. In $M_2(\mathbb{R})$, the skew-symmetric subspace $U$ inside the traceless subspace $W$ admits $W^{\prime }=\mathrm{span}\{I,\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\}$.

For vector subspaces $U,W$ of a common vector space, this is a surjective linear map. Its kernel of a linear map is $\{(x,-x):x\in U\cap W\}$, which has a linear isomorphism to $U\cap W$ by $x\mapsto (x,-x)$. The rank-nullity theorem and the direct sum basis give the dimension formula for a sum of subspaces. The paired domain keeps the two contributions separate even when their underlying vectors overlap.

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. If a continuous linear map has a Hausdorff target, its kernel of a linear map is a closed vector subspace, since it is the preimage of the closed singleton $\{0\}$. The target condition matters because our topological vector space convention permits non-Hausdorff spaces.

For a counterexample without the target hypothesis, the identity linear map of $\mathbb{R}$ with the indiscrete topology is continuous, but its kernel of a linear map $\{0\}$ is not closed.

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 $W$.

The smallest affine subspace containing a nonempty set $S$. For any chosen $a\in S$, the displayed formula proves existence and minimality: every containing affine subspace must contain all the indicated differences in its direction space. Equivalently it consists of finite affine combinations ${\sum }_i{\lambda }_i{s}_i$ with ${\sum }_i{\lambda }_i=1$. Its dimension can be smaller than the dimension of an affine hyperplane in which the set has been embedded.

In a finite-dimensional real vector space, a level set of a nonzero linear functional is an affine hyperplane. Choosing $v_0$ in that level set gives $H=v_0+\ker \ell$, so its dimension is one less than that of the ambient space. With a Euclidean inner product, translation followed by an orthonormal basis of $\ker \ell$ identifies it isometrically with the lower-dimensional Euclidean space. The coordinate-sum constraint on balanced cut vectors is an example.

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.

Let $V$ be a vector space over a specified field $F$, and let $v_1,\dots ,v_n\in V$. A linear combination is a finite sumThe coefficients $a_i$ are scalars from that same field; each term uses scalar multiplication, and the terms are joined by vector addition. For coefficients in the real numbers, negative values reverse arrows and fractional values scale them by fractional amounts. The collection of attainable vectors is the linear span. The column picture in Strang's MIT notes interprets solving a linear system as choosing coefficients to reach a target vector.^[ref]

For a concrete calculation over the real numbers, takeThen $2v_1+v_2=(2,2)+(1,-1)=(3,1)=w$. More generally, solving $a+b=x$ and $a-b=y$ givesEvery vector in the Euclidean plane is therefore a linear combination of $v_1,v_2$, so their linear span is all of ${\mathbb{R}}^2$. The target $(3,1)$ has coefficients $(2,1)$; the target $(0,2)$ has coefficients $(1,-1)$, since $(1,1)-(1,-1)=(0,2)$. The displayed formula also makes the coefficients unique. Consequently $(v_1,v_2)$ is an ordered basis, with coordinate vector $[(x,y){]}_{(v_1,v_2)}=((x+y)/2,(x-y)/2{)}^T$.

The payoff is a reusable language for synthesis: linear combinations build vectors from chosen ingredients, linear span decides which targets are reachable, and linear independence decides whether the coefficients are unique. The ingredients can also be polynomials or functions, as long as they belong to a vector space over the specified field.

An integer linear combination is a linear combination whose coefficients are integers. The lattice basis definition uses unique integer linear combinations, rather than arbitrary real coefficients. For example, $(1,0)$ is a real linear combination of $(2,0)$ but is not an integer linear combination of that single vector.

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.

Vector addition is the vector space operation combining two vectors into their sum. It is commutative and associative, has the zero vector as identity, and distributes with scalar multiplication as specified by the vector space axioms. Coordinatewise addition describes it after choosing a basis.

The vector sum of two vectors $v,w$ is the vector $v+w$ obtained by vector addition.

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. In finite dimension this counts the scalars needed to specify an arbitrary vector once an ordered basis is chosen. The exchange argument behind the basis extension theorem shows that any two finite bases have the same size: an independent family cannot be larger than a spanning family.

Thus ${\mathbb{R}}^2$ has dimension two even with an oblique basis, $\mathbb{R}[x{]}_{\le 2}$ has dimension three, and the homogeneous plane in vector subspace has dimension two. This count concerns independent freedom, so an embedded plane remains two-dimensional while sitting in a three-dimensional ambient vector space. Dimension also depends on the field, as the two interpretations of $\mathbb{C}$ in vector space illustrate.

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.

The direct sum $U\oplus W$ of two vector spaces consists of pairs $(u,w)$ with componentwise operations. A vector space is an internal direct sum of subspaces $U$ and $W$ exactly when every vector has a unique expression $u+w$ with $u\in U$ and $w\in W$.

For bases $(u_i)$ of $U$ and $(v_j)$ of $W$, the displayed pairs form a basis of $U\oplus W$. Expanding each component proves spanning; comparing components of a vanishing linear combination proves linear independence. Therefore $\dim (U\oplus W)=\dim U+\dim W$ for two finite-dimensional vector spaces.

If $V=A_1\oplus A_2=A_2\oplus A_3=A_1\oplus A_3$ with nonzero subspaces, choose $0\ne v\in A_3$ and write $v=a_1+a_2$ using the first direct sum of vector spaces. Both components are nonzero. The plane spanned by them meets $A_1$ and $A_2$ in their component lines and meets $A_3$ in the line through $v$. Its intersection with $A_3$ cannot be the whole plane, since $a_1\notin A_3$.

An orthogonal direct sum is a direct sum whose distinct summands are mutually orthogonal. For $v\in V$ and $w\in W$, the Pythagorean identity gives $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2$. In a Hilbert space, a closed subspace of a Hilbert space and its orthogonal complement give an orthogonal direct sum of the entire space. The component maps are orthogonal projections and have operator norm one when their ranges are nonzero.

A submodule $N$ of a module $M$ is a direct summand if some submodule $L$ gives $M=N\oplus L$. Equivalently, the inclusion has a left inverse R-module homomorphism, or $N$ is the image of an idempotent endomorphism of $M$.

Integer row vectors forming a square unimodular matrix are a basis of the corresponding free abelian group: the adjugate formula makes the inverse integral. The subgroup spanned by any subset of that basis is therefore a direct summand, complemented by the remaining basis vectors. In contrast, a proper finite-index subgroup of a free abelian group cannot be a direct summand: its nonzero finite quotient would have to embed as a subgroup of a torsion-free group.

For finite-dimensional subspaces $U,W$ of a common vector space,

For every subspace $U$ of a finite-dimensional vector space $V$, extending a basis of $U$ to a basis of $V$ produces a subspace $W$ such that $V=U\oplus W$.

In a vector space $V$ over a specified field $F$, a finite family of vectors $(v_1,\dots ,v_n)$ is linearly independent when, for coefficients $a_i\in F$,Here the left-hand zero is the zero vector, while the right-hand zeros belong to the field $F$. Thus linear independence excludes every nontrivial linear relation. For an infinite family, require this condition for every finite subfamily.^[ref]

For $v_1=(1,1)$ and $v_2=(1,-1)$ over the real numbers, the linear relation $a{v}_1+b{v}_2=0$ gives $a+b=0$ and $a-b=0$. Adding and subtracting yields $2a=2b=0$, hence $a=b=0$. This verifies linear independence directly and explains why the linear combination giving $(3,1)$ has exactly the coefficients $(2,1)$.

A linearly independent family cannot contain the zero vector: a coefficient of $1$ on that entry and $0$ on every other entry would be a nontrivial linear relation. The empty family is linearly independent, because its empty sum is the zero vector and there are no coefficients that could be nonzero. Linear independence identifies a collection without redundant vectors; combined with a spanning set, it gives a basis. The uniqueness of linear combination coefficients makes its representations unambiguous.

In the one-sorted vector-space language naming each scalar multiplication, a first-order formula reduces by quantifier elimination to finitely many linear equations. An independent pair satisfies only the identically zero equations among them. A dependent pair $(x,ax)$ with nonzero $x$ satisfies the same listed equations if the scalar $a$ avoids their finitely many forbidden ratios, which is possible over an infinite field. Thus no formula defines pair independence. The complete independent-pair type is still consistent in an infinite-dimensional model; it is nonisolated.