Meta tags:

Headings (most frequently used words):

and, spaces, vector, references, product, space, basic, subspaces, linear, matrices, numbers, direct, contents, definition, properties, bases, coordinates, history, examples, maps, constructions, with, additional, structure, related, structures, notes, citations, external, links, arrows, in, the, plane, ordered, pairs, of, coordinate, complex, other, field, extensions, function, equations, eigenvalues, eigenvectors, quotient, sum, tensor, normed, inner, topological, algebras, over, fields, bundles, modules, affine, projective, algebra, analysis, historical, further, banach, hilbert,

Text of the page (most frequently used words):
the (529), displaystyle (259), vector (252), and (229), mathbf (184), space (157), spaces (134), #linear (114), that (97), are (91), for (76), with (69), algebra (62), 978 (59), isbn (57), this (52), called (50), multiplication (47), matrix (46), scalar (45), vectors (43), product (43), field (43), edit (39), two (38), such (38), example (36), which (36), basis (34), addition (34), functions (30), new (29), also (29), over (29), sum (29), elements (28), not (28), any (28), given (28), mathematics (27), map (27), lang (27), can (27), subspace (26), dimensional (26), function (26), york (25), from (24), numbers (24), set (24), otimes (24), theorem (23), springer (23), topological (22), dimension (22), real (22), one (22), all (22), maps (22), analysis (21), same (21), equations (20), main (20), group (19), its (19), hilbert (19), affine (18), where (18), theory (17), form (17), 2005 (17), 1987 (17), bundle (17), ring (17), infinite (17), cdot (17), geometry (16), their (16), verlag (16), finite (16), complex (16), roman (15), only (15), there (15), norm (15), they (15), infty (15), right (15), matrices (14), berlin (14), axioms (14), case (14), subspaces (14), isomorphism (14), ldots (14), coordinate (13), tensor (13), inner (13), has (13), differential (13), 1991 (13), bases (13), arrow (13), fixed (13), solutions (13), element (13), mathematical (12), other (12), equation (12), 387 (12), 1995 (12), article (12), every (12), but (12), left (12), structure (11), orthogonal (11), examples (11), kreyszig (11), 1989 (11), topology (11), some (11), banach (11), plane (11), line (11), between (11), many (11), modules (11), direct (11), articles (10), projective (10), span (10), decomposition (10), polynomial (10), press (10), doi (10), 1999 (10), bourbaki (10), definition (10), omega (10), complete (10), since (10), more (10), denoted (10), above (10), have (10), related (10), sequence (10), prime (10), arrows (10), coordinates (10), terms (9), using (9), different (9), advanced (9), algebras (9), john (9), wiley (9), tangent (9), algebraic (9), known (9), whose (9), isomorphic (9), ordered (9), number (9), scalars (9), toggle (8), additional (8), may (8), image (8), physics (8), structures (8), basic (8), functional (8), combination (8), introduction (8), lie (8), series (8), sons (8), bundles (8), general (8), via (8), 1993 (8), 2002 (8), lebesgue (8), zero (8), property (8), particular (8), consists (8), defined (8), properties (8), continuous (8), langle (8), rangle (8), lim (8), first (8), corresponding (8), written (8), pair (8), wikipedia (7), under (7), page (7), normed (7), identity (7), operations (7), determinant (7), vol (7), standard (7), hall (7), independent (7), implies (7), references (7), peano (7), 2000 (7), abstract (7), applied (7), applications (7), brown (7), see (7), when (7), blue (7), into (7), like (7), way (7), cdots (7), another (7), denotes (7), polynomials (7), direction (7), text (6), german (6), category (6), point (6), transformation (6), euclidean (6), system (6), geometric (6), equivalence (6), positive (6), 1998 (6), 540 (6), american (6), texts (6), commutative (6), grassmann (6), möbius (6), what (6), 1967 (6), treves (6), second (6), them (6), operation (6), used (6), respectively (6), subset (6), mapsto (6), module (6), rings (6), allows (6), because (6), introduced (6), endowed (6), time (6), means (6), these (6), generally (6), exists (6), frac (6), convergence (6), coefficients (6), quotient (6), pairs (6), sin (6), exp (6), length (6), subsection (6), view (5), non (5), cs1 (5), french (5), numerical (5), free (5), tensors (5), eigenvalue (5), 2001 (5), 1994 (5), pdf (5), groups (5), 2nd (5), david (5), erwin (5), engineering (5), axiom (5), choice (5), study (5), nicolas (5), der (5), history (5), integral (5), trivial (5), fields (5), rise (5), contains (5), associative (5), common (5), objects (5), then (5), important (5), homogeneous (5), times (5), red (5), abelian (5), certain (5), nonzero (5), equipped (5), much (5), variables (5), bilinear (5), eigenvalues (5), converges (5), approximation (5), combinations (5), key (5), considering (5), fundamental (5), unit (5), sums (5), negative (5), textstyle (5), begin (5), end (5), constructions (5), kernel (5), linearly (5), expressed (5), equivalently (5), eigenvectors (5), systems (5), contents (4), search (4), about (4), agree (4), was (4), concepts (4), index (4), symmetric (4), dual (4), hamel (4), least (4), elementary (4), order (4), dot (4), inverse (4), hamilton (4), links (4), oclc (4), 1974 (4), prentice (4), volume (4), michael (4), 1007 (4), 486 (4), dover (4), 471 (4), william (4), calculus (4), 1966 (4), society (4), halmos (4), graduate (4), 8218 (4), existence (4), 1969 (4), atiyah (4), fourier (4), 2011 (4), serge (4), partial (4), integration (4), nicholson (4), 2018 (4), iii (4), section (4), integrable (4), shows (4), leq (4), gives (4), relation (4), does (4), roughly (4), instead (4), locally (4), points (4), however (4), products (4), yields (4), operatorname (4), include (4), operator (4), physical (4), often (4), limit (4), similar (4), commonly (4), sense (4), together (4), int (4), satisfying (4), following (4), ones (4), sup (4), consisting (4), notions (4), notion (4), specifying (4), three (4), too (4), difference (4), therefore (4), follows (4), rules (4), shown (4), alignedat (4), simply (4), tuples (4), ker (4), arbitrary (4), intersection (4), containing (4), aligned (4), respect (4), extensions (4), binary (4), lattice (4), hide (4), move (4), sidebar (4), topic (3), statement (3), use (3), categories (3), italian (3), language (3), sources (3), images (3), description (3), data (3), cartesian (3), component (3), treatment (3), multilinear (3), projection (3), nullity (3), rank (3), row (3), mapping (3), independence (3), computations (3), value (3), spectral (3), definite (3), canonical (3), jordan (3), eigenvector (3), cauchy (3), encyclopedia (3), university (3), 2007 (3), issn (3), still (3), chapman (3), publications (3), minkowski (3), mac (3), lane (3), 3rd (3), 1996 (3), macdonald (3), providence (3), classics (3), addison (3), wesley (3), 201 (3), further (3), 1888 (3), moore (3), 1827 (3), barycentric (3), quaternions (3), exactly (3), hermann (3), dorier (3), 1833 (3), bellavitis (3), 1804 (3), bolzano (3), academic (3), wolff (3), schaefer (3), crc (3), harmonic (3), brooks (3), cole (3), choquet (3), 8176 (3), meyer (3), third (3), 2020 (3), xvi (3), corollary (3), being (3), riemann (3), provided (3), context (3), uniform (3), considered (3), contain (3), especially (3), precisely (3), itself (3), compared (3), those (3), division (3), differentiable (3), manifold (3), forms (3), information (3), contrast (3), family (3), dim (3), makes (3), seen (3), symbols (3), simple (3), crucial (3), defining (3), associated (3), limits (3), cardinality (3), closure (3), belonging (3), interval (3), domain (3), mathbb (3), topologies (3), functionals (3), norms (3), each (3), here (3), changes (3), special (3), lengths (3), componentwise (3), part (3), completely (3), whether (3), added (3), universal (3), say (3), concept (3), closed (3), origin (3), equivalent (3), det (3), moreover (3), represented (3), natural (3), said (3), multiples (3), satisfy (3), occur (3), well (3), around (3), distributivity (3), endomorphism (3), tools (3), languages (2), table (2), contact (2), privacy (2), policy (2), apply (2), wikimedia (2), commons (2), maint (2), location (2), missing (2), publisher (2), pages (2), multiple (2), scaled (2), short (2), wikidata (2), retrieved (2), states (2), sparse (2), arithmetic (2), exterior (2), classical (2), orientation (2), null (2), complement (2), change (2), gram (2), schmidt (2), process (2), squares (2), relations (2), formula (2), schur (2), higher (2), singular (2), diagonal (2), cayley (2), eigenspace (2), spectrum (2), characteristic (2), strassen (2), elimination (2), classes (2), outline (2), ems (2), wikibook (2), external (2), cambridge (2), studies (2), charles (2), 1992 (2), archived (2), original (2), varadarajan (2), london (2), 412 (2), ian (2), spivak (2), computing (2), naber (2), gregory (2), 2003 (2), mathematician (2), saunders (2), husemoller (2), single (2), 2035388 (2), jstor (2), 2307 (2), paul (2), upper (2), quantum (2), mechanics (2), griffiths (2), eisenbud (2), eisenberg (2), guy (2), 1979 (2), proof (2), hairy (2), ball (2), 2320587 (2), coxeter (2), chapters (2), blass (2), 1984 (2), book (2), francis (2), link (2), ausdehnungslehre (2), axiomatization (2), 1006 (2), hmat (2), historia (2), mathematica (2), ein (2), analytic (2), 1853 (2), guo (2), 2021 (2), kannenberg (2), extension (2), 1844 (2), jean (2), 1922 (2), 181 (2), 133 (2), les (2), application (2), sets (2), stefan (2), historical (2), boston (2), kernels (2), rudin (2), lawrence (2), 1983 (2), steven (2), 534 (2), evans (2), princeton (2), dudley (2), wadsworth (2), dennery (2), krzywicki (2), 2004 (2), stoll (2), wong (2), 1968 (2), 8247 (2), 135 (2), 89871 (2), 4757 (2), 1949 (2), joshi (2), jain (2), heil (2), 4687 (2), 1948 (2), grillet (2), pierre (2), 71568 (2), artin (2), anton (2), rorres (2), 2010 (2), version (2), chapter (2), lemma (2), 106 (2), algèbre (2), citations (2), sections (2), would (2), get (2), everywhere (2), inequality (2), authors (2), concrete (2), shape (2), while (2), associativity (2), confused (2), specific (2), produces (2), top (2), method (2), notes (2), lines (2), generalize (2), specified (2), translating (2), yield (2), need (2), development (2), counterpart (2), multiplicative (2), cotangent (2), provide (2), underlying (2), parametrized (2), circle (2), globally (2), consequences (2), required (2), depending (2), large (2), latter (2), whereas (2), identifying (2), strip (2), formal (2), spanned (2), imposing (2), forcing (2), equal (2), leads (2), law (2), several (2), possess (2), hyperbola (2), various (2), values (2), correspond (2), wavefunctions (2), conversely (2), approximated (2), closely (2), sufficient (2), weierstrass (2), overline (2), summation (2), sine (2), derivatives (2), than (2), sequences (2), inequivalent (2), components (2), ell (2), qquad (2), ensure (2), necessary (2), give (2), without (2), spheres (2), consider (2), depends (2), cases (2), pointwise (2), compatible (2), talk (2), amount (2), reals (2), angle (2), variant (2), lorentz (2), out (2), opposed (2), dimensions (2), useful (2), cos (2), done (2), datum (2), measures (2), sqrt (2), express (2), characterized (2), framework (2), deal (2), question (2), likewise (2), finitely (2), unique (2), diagram (2), define (2), composition (2), tuple (2), both (2), construction (2), allowed (2), preserving (2), statements (2), very (2), assignment (2), less (2), smallest (2), planes (2), particularly (2), uniquely (2), vice (2), versa (2), correspondence (2), injective (2), down (2), back (2), essentially (2), exponential (2), bmatrix (2), triples (2), thus (2), belongs (2), class (2), mentioned (2), simplest (2), viewed (2), reduces (2), dots (2), stretched (2), opposite (2), pointing (2), forces (2), black (2), his (2), work (2), reasonings (2), says (2), empty (2), satisfied (2), concise (2), four (2), remaining (2), homomorphism (2), eight (2), must (2), assigns (2), below (2), areas (2), kinds (2), appearance (2), upload (2), file (2), read (2), english (2), norsk (2), bahasa (2), log (2), create (2), account (2), donate (2), menu (2), add, mobile, cookie, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, site, you, registered, trademark, profit, organization, foundation, inc, creative, attribution, sharealike, license, rendered, parsoid, last, edited, september, 2026, utc, hidden, good, auto, https, org, php, title, vector_space, oldid, 1373258776, yale, lux, israel, czech, republic, bnf, france, united, national, gnd, international, authority, control, databases, comparison, libraries, subprograms, stability, floating, quadric, galilean, poincaré, flat, gamas, multivector, bivector, clifford, outer, symplectic, improper, rotation, indefinite, pseudo, orthogonality, cyclic, column, squeeze, shear, woodbury, householder, consimilarity, similarity, congruence, reducing, haynsworth, inertia, additivity, polar, cholesky, decompositions, list, hankel, circulant, toeplitz, stochastic, vandermonde, hessian, hessenberg, block, tridiagonal, triangular, perron, frobenius, pfaffian, semi, antihermitian, hermitian, unitary, skew, transpose, adjugate, pseudoinverse, weyr, normal, trace, algorithm, overcompleteness, gauss, gaussian, cramer, rule, binet, minor, template, glossary, 36131259, 1269324, 521, 55987, homological, weibel, wallace, feb, xxxiv, 2017, 0098, 3063, 1109, 125072, xviii, ieee, transactions, consumer, electronics, jpeg, picture, compression, 535732, representations, 1975, 10800, galois, stewart, houston, publish, perish, comprehensive, 1971, schnelle, multiplikation, großer, zahlen, fast, big, 292, 9738629, s2cid, 0010, 485x, bf02242355, 281, volker, schönhage, 2044239, 43235, spacetime, 1973, freeman, 7167, 0344, gravitation, wheeler, archibald, thorne, kip, misner, 98403, working, luenberger, 1997, 18117, optimization, methods, 8th, 15496, xiv, 352, 66721, 4th, 25907, riemannian, jost, jürgen, dale, 94087, fibre, hughes, hallett, deborah, mccallum, gleason, andrew, 2013, 0470, 88861, multivariable, halpern, james, jun, 673, 670, proceedings, 90093, saddle, river, 124405, goldrei, derek, 1st, 60610, classic, guided, 150, 1322960, 94269, murray, robert, 574, 572, monthly, 96532, harold, scott, 64241, 64243, andreas, contemporary, 0763890, 5026, axiomatic, grant, 1043170, 09394, 1976, toronto, thomson, learning, 083993, solid, state, mermin, ashcroft, neil, 1901, internet, archive, vct, formulario, mathematico, citation, turin, calcolo, geometrico, secondo, preceduto, dalle, operazioni, della, logica, deduttiva, giuseppe, 1875, 1940, 303, 1025, 262, 2006, barycentrische, calcul, neues, hülfsmittel, zur, analytischen, behandlung, geometrie, utility, august, ferdinand, royal, irish, academy, lectures, rowan, hongyu, world, scientific, 981, 4103, reprint, translated, lloyd, 2031, wigand, die, lineale, neuer, zweig, mathematik, 1822, chez, firmin, didot, père, fils, théorie, analytique, chaleur, baptiste, joseph, luc, 261, 1347828, 1024, 227, genesis, paris, éléments, histoire, des, mathématiques, sopra, alcune, applicazioni, nuovo, metodo, geometria, analitica, verona, poligrafo, giornale, scienze, lettre, arti, giuso, betrachtungen, über, einige, gegenstände, elementargeometrie, considerations, aspects, bernard, 0016, 2736, 4064, fundamenta, mathematicae, sur, opérations, dans, ensembles, abstraits, leur, aux, équations, intégrales, distributions, françois, manfred, imprint, 840278135, 4612, 7155, gtm, helmut, walter, mcgraw, hill, 0070542368, narici, beckenstein, edward, pure, boca, raton, 144216834, 1584888666, loomis, lynn, 1953, 702357363, 48123, 2027, uc1, b4250788, hdl, 94001, 14179, library, 0992618, 50459, introductory, 1988, 6th, 85824, krantz, carus, monographs, washington, association, america, 88385, 031, panorama, ifeachor, emmanuel, jervis, barrie, harlow, essex, england, published, 59619, digital, signal, processing, practical, approach, gasquet, claude, witomski, patrick, 98485, filtering, computation, wavelets, folland, gerald, 17094, 0772, dunham, 691, 09565, gallery, richard, pacific, grove, books, software, 10050, probability, philippe, andre, courier, 69193, physicists, gustave, bse, braun, martin, 97894, 41129, 13627, 9th, 56799, van, waerden, bartel, leendert, 4840, spindler, karlheinz, 9144, 24766, keith, lyryx, carl, 454, siam, soc, 222, 1646, 193, 211, revised, 1878556, 95385, undergraduate, 119, 45592, foundations, discrete, 8493, 0978, christopher, birkhäuser, primer, expanded, edition, antoine, 242, science, business, media, dekker, 8419, 510, howard, chris, 10th, exercise, 442, 436, 108, 121, xvii, xiii, 349, 190, 102, cor, proposition, 204, 205, yoneda, 140, 156, vii, 198, xii, 335, 358, 359, 355, 268, 271, linéaire, multilinéaire, 126, 450, 212, 101, 185, vanishes, nowhere, just, restricts, isomorphisms, fibers, homeomorphism, thing, distinction, neq, measure, unbounded, cannot, integrated, advantages, 125, technical, reasons, identify, seminorm, almost, triangle, requirement, loc, choose, start, derive, typically, necessarily, refers, assumed, denote, typographical, distinguishing, vec, formalize, idea, intersecting, infinity, parametrizing, flags, flag, manifolds, grassmannians, parallel, origins, obtained, inhomogeneous, generalizing, discussed, found, setting, solution, nullspace, coset, action, transitive, light, shifted, complicated, presence, including, nevertheless, compactly, term, mean, algebro, interpretation, inverses, collection, global, deepening, geometrical, insight, purely, classification, octonions, sphere, continuously, restriction, despite, character, twisted, former, orientable, cylinder, open, intervals, neighborhood, fiber, looks, adding, obtain, varies, concatenating, requiring, commute, distributive, degree, cross, commutator, jacobi, anticommutativity, neither, nor, failure, limited, constraints, again, quotients, interpreted, great, lead, frequently, dependent, describes, energy, momentum, decomposes, acting, eigenfunctions, compact, eigenstates, schrödinger, finding, desirable, approximate, equally, early, guise, established, desired, technique, conceptually, suffice, generate, whole, exhibit, suitable, purposes, enables, construct, generalization, axes, expansion, trigonometric, stone, taylor, honor, conjugate, succeeding, snapshots, show, approximating, periodic, boundedness, conditions, sobolev, uses, justification, concretely, condition, replaces, entries, conceptual, should, match, concerned, separating, appropriate, hahn, restrict, attention, uniformly, piece, focuses, balls, enclose, will, richer, consist, depicted, bigger, diamond, depicts, could, imposed, prominent, mode, questions, treated, carrying, vary, bounded, make, carry, close, reflects, takes, singling, fourth, note, conventions, zeroeth, relativity, cosines, measuring, angles, entails, comparing, relies, ability, riesz, understood, insofar, offer, adapted, needs, require, per, recipient, subject, universality, dotted, equals, instance, indirectly, object, depicting, central, deals, extending, specify, performed, coprod, coproduct, bigoplus, prod, ways, combining, indexed, corpus, behaves, formulated, proven, equiv, extends, appear, differentiation, procedure, constant, mapped, lies, forgets, contained, modulo, subseteq, referred, hyperplane, nonempty, ambient, unambiguously, own, passing, through, thick, green, yellow, rephrased, determinants, having, spelling, expression, hand, side, enough, automatically, happens, phenomenon, governed, eigenbasis, algebraically, lambda, endomorphisms, tells, square, absolute, formed, parallelepiped, after, choosing, encode, rectangular, array, typical, once, chosen, determined, preferred, classified, embedded, bidual, hom, planar, departing, going, turns, describing, possible, onto, identical, identities, holding, transported, surjective, compositions, reflect, preserve, constants, bxe, vein, tied, ratios, condense, namely, performing, similarly, situations, behaved, linearity, possessing, differentiability, integrability, continuity, subsets, smaller, usual, follow, fact, hold, think, representing, earlier, imaginary, usually, regarded, previous, endpoint, significant, few, resulting, double, starting, describe, starts, scaling, dilated, shrunk, multiplying, parallelogram, velocities, due, later, formalized, 1920, began, interact, notably, henri, 1857, harmonization, simplification, studied, initiated, envisaged, present, exceeds, led, him, today, modern, although, did, develop, 1897, adopted, made, initial, inroads, salvatore, pincherle, notation, were, reconsidered, presentation, inception, treating, goes, 1867, who, laguerre, argand, stem, 1636, mathematicians, founded, achieve, predecessors, directed, segments, share, equipollence, curve, fermat, rené, descartes, tool, base, explicitly, described, rational, detailed, remainder, generating, spanning, generates, spans, belong, induced, results, even, concisely, subtraction, defines, specifically, establish, most, compatibility, additive, commutativity, listed, boldface, distinguish, hopf, bialgebra, graded, operators, lattices, boolean, heyting, total, complemented, semilattice, near, semiring, rng, magma, quasigroup, loop, rack, quandle, monoid, semigroup, speaking, specifies, directions, depend, otherwise, naturally, continuum, countably, cardinal, allow, modeling, provides, synthetic, manipulating, studying, magnitude, velocity, quantities, multiplied, requirements, based, illustration, factor, yielding, redirects, incidence, item, wikiversity, wikibooks, projects, printable, download, print, export, switch, legacy, parser, shortened, url, cite, permanent, actions, 閩南語, bân, lâm, tiếng, việt, vèneto, اردو, українська, türkçe, tagalog, ไทย, தமிழ், svenska, српски, srpski, shqip, slovenščina, slovenčina, srpskohrvatski, српскохрватски, sicilianu, русский, română, português, پنجابی, piemontèis, polski, ਪੰਜਾਬੀ, occitan, bokmål, nynorsk, nederlands, မြန်မာဘာသာ, melayu, മലയാളം, македонски, मगही, latviešu, lietuvių, ລາວ, lombard, latina, кыргызча, 한국어

Text of the page (random words):
considered with the presentation of complex numbers by argand and hamilton and the inception of quaternions by the latter 23 they are elements in r 2 and r 4 treating them using linear combinations goes back to laguerre in 1867 who also defined systems of linear equations in 1857 cayley introduced the matrix notation which allows for harmonization and simplification of linear maps around the same time grassmann studied the barycentric calculus initiated by möbius he envisaged sets of abstract objects endowed with operations 24 in his work the concepts of linear independence and dimension as well as scalar products are present grassmann s 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called algebras italian mathematician peano was the first to give the modern definition of vector spaces and linear maps in 1888 25 although he called them linear systems 26 peano s axiomatization allowed for vector spaces with infinite dimension but peano did not develop that theory further in 1897 salvatore pincherle adopted peano s axioms and made initial inroads into the theory of infinite dimensional vector spaces 27 an important development of vector spaces is due to the construction of function spaces by henri lebesgue this was later formalized by banach and hilbert around 1920 28 at that time algebra and the new field of functional analysis began to interact notably with key concepts such as spaces of p integrable functions and hilbert spaces 29 examples edit main article examples of vector spaces arrows in the plane edit vector addition the sum v w black of the vectors v blue and w red is shown scalar multiplication the multiples v and 2 w are shown the first example of a vector space consists of arrows in a fixed plane starting at one fixed point this is used in physics to describe forces or velocities 30 given any two such arrows v and w the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin too this new arrow is called the sum of the two arrows and is denoted v w in the special case of two arrows on the same line their sum is the arrow on this line whose length is the sum or the difference of the lengths depending on whether the arrows have the same direction another operation that can be done with arrows is scaling given any positive real number a the arrow that has the same direction as v but is dilated or shrunk by multiplying its length by a is called multiplication of v by a it is denoted a v when a is negative a v is defined as the arrow pointing in the opposite direction instead 31 the following shows a few examples if a 2 the resulting vector a w has the same direction as w but is stretched to the double length of w the second image equivalently 2 w is the sum w w moreover 1 v v has the opposite direction and the same length as v blue vector pointing down in the second image ordered pairs of numbers edit a second key example of a vector space is provided by pairs of real numbers x and y the order of the components x and y is significant so such a pair is also called an ordered pair such a pair is written as x y the sum of two such pairs and the multiplication of a pair with a number is defined as follows 32 x 1 y 1 x 2 y 2 x 1 x 2 y 1 y 2 a x y a x a y displaystyle begin aligned x_ 1 y_ 1 x_ 2 y_ 2 x_ 1 x_ 2 y_ 1 y_ 2 a x y ax ay end aligned the first example above reduces to this example if an arrow is represented by a pair of cartesian coordinates of its endpoint coordinate space edit the simplest example of a vector space over a field f is the field f itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication more generally all n tuples sequences of length n a 1 a 2 a n displaystyle a_ 1 a_ 2 dots a_ n of elements a i of f form a vector space that is usually denoted f n and called a coordinate space 33 the case n 1 is the above mentioned simplest example in which the field f is also regarded as a vector space over itself the case f r and n 2 so r 2 reduces to the previous example complex numbers and other field extensions edit the set of complex numbers c numbers that can be written in the form x iy for real numbers x and y where i is the imaginary unit form a vector space over the reals with the usual addition and multiplication x iy a ib x a i y b and c x iy c x i c y for real numbers x y a b and c the various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic the example of complex numbers is essentially the same as that is it is isomorphic to the vector space of ordered pairs of real numbers mentioned above if we think of the complex number x i y as representing the ordered pair x y in the complex plane then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example more generally field extensions provide another class of examples of vector spaces particularly in algebra and algebraic number theory a field f containing a smaller field e is an e vector space by the given multiplication and addition operations of f 34 for example the complex numbers are a vector space over r and the field extension q i 5 displaystyle mathbf q i sqrt 5 is a vector space over q function spaces edit main article function space addition of functions the sum of the sine and the exponential function is sin exp r r displaystyle sin exp mathbb r to mathbb r with sin exp x sin x exp x displaystyle sin exp x sin x exp x functions from any fixed set ω to a field f also form vector spaces by performing addition and scalar multiplication pointwise that is the sum of two functions f and g is the function f g displaystyle f g given by f g w f w g w displaystyle f g w f w g w and similarly for multiplication such function spaces occur in many geometric situations when ω is the real line or an interval or other subsets of r many notions in topology and analysis such as continuity integrability or differentiability are well behaved with respect to linearity sums and scalar multiples of functions possessing such a property still have that property 35 therefore the set of such functions are vector spaces whose study belongs to functional analysis linear equations edit main articles linear equation linear differential equation and systems of linear equations systems of homogeneous linear equations are closely tied to vector spaces 36 for example the solutions of a 3 b c 0 4 a 2 b 2 c 0 displaystyle begin alignedat 9 a 3b c 0 4 a 2b 2 c 0 end alignedat are given by triples with arbitrary a displaystyle a b a 2 displaystyle b a 2 and c 5 a 2 displaystyle c 5a 2 they form a vector space sums and scalar multiples of such triples still satisfy the same ratios of the three variables thus they are solutions too matrices can be used to condense multiple linear equations as above into one vector equation namely a x 0 displaystyle a mathbf x mathbf 0 where a 1 3 1 4 2 2 displaystyle a begin bmatrix 1 3 1 4 2 2 end bmatrix is the matrix containing the coefficients of the given equations x displaystyle mathbf x is the vector a b c displaystyle a b c a x displaystyle a mathbf x denotes the matrix product and 0 0 0 displaystyle mathbf 0 0 0 is the zero vector in a similar vein the solutions of homogeneous linear differential equations form vector spaces for example f x 2 f x f x 0 displaystyle f prime prime x 2f prime x f x 0 yields f x a e x b x e x displaystyle f x ae x bxe x where a displaystyle a and b displaystyle b are arbitrary constants and e x displaystyle e x is the natural exponential function linear maps and matrices edit main article linear map the relation of two vector spaces can be expressed by linear map or linear transformation they are functions that reflect the vector space structure that is they preserve sums and scalar multiplication f v w f v f w f a v a f v displaystyle begin aligned f mathbf v mathbf w f mathbf v f mathbf w f a cdot mathbf v a cdot f mathbf v end aligned for all v displaystyle mathbf v and w displaystyle mathbf w in v displaystyle v all a displaystyle a in f displaystyle f 37 an isomorphism is a linear map f v w such that there exists an inverse map g w v which is a map such that the two possible compositions f g w w and g f v v are identity maps equivalently f is both one to one injective and onto surjective 38 if there exists an isomorphism between v and w the two spaces are said to be isomorphic they are then essentially identical as vector spaces since all identities holding in v are via f transported to similar ones in w and vice versa via g describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces for example the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above see examples are isomorphic a planar arrow v departing at the origin of some fixed coordinate system can be expressed as an ordered pair by considering the x and y component of the arrow as shown in the image at the right conversely given a pair x y the arrow going by x to the right or to the left if x is negative and y up down if y is negative turns back the arrow v 39 linear maps v w between two vector spaces form a vector space hom f v w also denoted l v w or 𝓛 v w 40 the space of linear maps from v to f is called the dual vector space denoted v 41 via the injective natural map v v any vector space can be embedded into its bidual the map is an isomorphism if and only if the space is finite dimensional 42 once a basis of v is chosen linear maps f v w are completely determined by specifying the images of the basis vectors because any element of v is expressed uniquely as a linear combination of them 43 if dim v dim w a 1 to 1 correspondence between fixed bases of v and w gives rise to a linear map that maps any basis element of v to the corresponding basis element of w it is an isomorphism by its very definition 44 therefore two vector spaces over a given field are isomorphic if their dimensions agree and vice versa another way to express this is that any vector space over a given field is completely classified up to isomorphism by its dimension a single number in particular any n dimensional f vector space v is isomorphic to f n however there is no canonical or preferred isomorphism an isomorphism φ f n v is equivalent to the choice of a basis of v by mapping the standard basis of f n to v via φ matrices edit main articles matrix and determinant a typical matrix matrices are a useful notion to encode linear maps 45 they are written as a rectangular array of scalars as in the image at the right any m by n matrix a displaystyle a gives rise to a linear map from f n to f m by the following x x 1 x 2 x n j 1 n a 1 j x j j 1 n a 2 j x j j 1 n a m j x j displaystyle mathbf x x_ 1 x_ 2 ldots x_ n mapsto left sum _ j 1 n a_ 1j x_ j sum _ j 1 n a_ 2j x_ j ldots sum _ j 1 n a_ mj x_ j right where textstyle sum denotes summation or by using the matrix multiplication of the matrix a displaystyle a with the coordinate vector x displaystyle mathbf x x a x displaystyle mathbf x mapsto a mathbf x moreover after choosing bases of v and w any linear map f v w is uniquely represented by a matrix via this assignment 46 the volume of this parallelepiped is the absolute value of the determinant of the 3 by 3 matrix formed by the vectors r 1 r 2 and r 3 the determinant det a of a square matrix a is a scalar that tells whether the associated map is an isomorphism or not to be so it is sufficient and necessary that the determinant is nonzero 47 the linear transformation of r n corresponding to a real n by n matrix is orientation preserving if and only if its determinant is positive eigenvalues and eigenvectors edit main article eigenvalues and eigenvectors endomorphisms linear maps f v v are particularly important since in this case vectors v can be compared with their image under f f v any nonzero vector v satisfying λ v f v where λ is a scalar is called an eigenvector of f with eigenvalue λ 48 equivalently v is an element of the kernel of the difference f λ id where id is the identity map v v if v is finite dimensional this can be rephrased using determinants f having eigenvalue λ is equivalent to det f λ id 0 displaystyle det f lambda cdot operatorname id 0 by spelling out the definition of the determinant the expression on the left hand side can be seen to be a polynomial function in λ called the characteristic polynomial of f 49 if the field f is large enough to contain a zero of this polynomial which automatically happens for f algebraically closed such as f c any linear map has at least one eigenvector the vector space v may or may not possess an eigenbasis a basis consisting of eigenvectors this phenomenon is governed by the jordan canonical form of the map 50 the set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue and f in question basic constructions edit in addition to the above concrete examples there are a number of standard linear algebraic constructions that yield vector spaces related to given ones subspaces and quotient spaces edit main articles linear subspace and quotient vector space a line passing through the origin blue thick in r 3 is a linear subspace it is the intersection of two planes green and yellow a nonempty subset w displaystyle w of a vector space v displaystyle v that is closed under addition and scalar multiplication and therefore contains the 0 displaystyle mathbf 0 vector of v displaystyle v is called a linear subspace of v displaystyle v or simply a subspace of v displaystyle v when the ambient space is unambiguously a vector space 51 nb 4 subspaces of v displaystyle v are vector spaces over the same field in their own right the intersection of all subspaces containing a given set s displaystyle s of vectors is called its span and it is the smallest subspace of v displaystyle v containing the set s displaystyle s expressed in terms of elements the span is the subspace consisting of all the linear combinations of elements of s displaystyle s 52 linear subspace of dimension 1 and 2 are referred to as a line also vector line and a plane respectively if w is an n dimensional vector space any subspace of dimension 1 less i e of dimension n 1 displaystyle n 1 is called a hyperplane 53 the counterpart to subspaces are quotient vector spaces 54 given any subspace w v displaystyle w subseteq v the quotient space v w displaystyle v w v displaystyle v modulo w displaystyle w is defined as follows as a set it consists of v w v w w w displaystyle mathbf v w mathbf v mathbf w mathbf w in w where v displaystyle mathbf v is an arbitrary vector in v displaystyle v the sum of two such elements v 1 w displaystyle mathbf v _ 1 w and v 2 w displaystyle mathbf v _ 2 w is v 1 v 2 w displaystyle left mathbf v _ 1 mathbf v _ 2 r...