458:
4468:
4732:
999:
3698:. Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by
2831:
3840:
is a noncommutative ring, then the concept still generalizes, with one caveat: since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module. This is simply a matter of doing
646:
2507:
2987:
291:
3742:
The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a
3696:
3817:
when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis. The articles on the various flavors of topological vector spaces go into more detail about these.
313:
always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of
2671:
3734:
From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that
994:{\displaystyle {\begin{aligned}(a_{1},a_{2},a_{3})&=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\\&=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\\&=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}.\end{aligned}}}
296:
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of
3497:
All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.
3957:
1946:
1776:
2071:
1198:
651:
2235:
2303:
1625:
3106:
2308:
2869:
1289:
173:
3829:
instead of a field, then everything that has been said above about linear combinations generalizes to this case without change. The only difference is that we call spaces like this
3547:
3479:
Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an
3043:
3604:
3725:
3245:
3461:
of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone.
3400:
3341:
3436:
3289:
487:
3609:
2826:{\displaystyle \operatorname {span} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}):=\{a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}:a_{1},\ldots ,a_{n}\in K\}.}
2543:
326:(as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each
4298:
4326:
3872:
1791:
4717:
1636:
4253:
4231:
4212:
4186:
3553:, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector
1961:
1116:
2502:{\displaystyle {\begin{aligned}&0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\\={}&1x^{3}+0x^{2}+0x+(-1).\end{aligned}}}
335:; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.
2107:
509:
1506:
4170:
3051:
4707:
4669:
4605:
31:
4204:
2982:{\displaystyle \mathbf {v} =\sum _{i}a_{i}\mathbf {v} _{i}=\sum _{i}b_{i}\mathbf {v} _{i}{\text{ where }}a_{i}\neq b_{i}.}
1475:? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector
4447:
4319:
470:
4552:
4402:
3468:
are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and
480:
474:
466:
3731:
is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
1206:
4457:
4351:
2077:
4697:
4346:
3350:
1090:
4572:
3750:
Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.
491:
286:{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+a_{3}\mathbf {v} _{3}+\cdots +a_{n}\mathbf {v} _{n}.}
4689:
3763:
3465:
3464:
These concepts often arise when one can take certain linear combinations of objects, but not any: for example,
50:
385:, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set
2101:
is also −1. Therefore, the only possible way to get a linear combination is with these coefficients. Indeed,
4756:
4735:
4442:
4312:
3523:
3354:
428:; each individual linear combination will only involve finitely many vectors. Also, there is no reason that
3813: + ⋯, going on forever. Such infinite linear combinations do not always make sense; we call them
85:
and related fields of mathematics. Most of this article deals with linear combinations in the context of a
4499:
4432:
4422:
3161:
3457:
operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are
4659:
4514:
4509:
4504:
4437:
4382:
3550:
3517:
3491:
2995:
3556:
3701:
3221:
4524:
4489:
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4367:
3833:
127:
57:
of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of
3176:
By restricting the coefficients used in linear combinations, one can define the related concepts of
4702:
4582:
4557:
4407:
3306:
3181:
2842:
1369:
1036:
90:
346:
may be specified explicitly, or they may be obvious from context. In that case, we often speak of
4196:
3372:
3365:
3313:
3251:
3185:
3177:
3129:
3691:{\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots }
3490:
If one allows only scalar multiplication, not addition, one obtains a (not necessarily convex)
4610:
4567:
4494:
4387:
4249:
4227:
4208:
4182:
3727:
being or the standard simplex being model spaces, and such observations as that every bounded
3405:
3258:
315:
54:
4615:
4519:
4372:
4174:
3826:
3469:
584:
17:
4674:
4467:
4427:
4417:
3728:
3494:; one often restricts the definition to only allowing multiplication by positive scalars.
3293:
3214:
544:
436:; in that case, we declare by convention that the result of the linear combination is the
2522:
322:
of vectors is linearly independent precisely if any linear combination of the vectors in
2863:, a single vector can be written in two different ways as a linear combination of them:
4679:
4664:
4600:
4335:
4241:
3744:
3473:
1952:
1098:
1018:
82:
4750:
4712:
4635:
4595:
4562:
4542:
4162:
3513:
3507:
3480:
2512:
However, when we set corresponding coefficients equal in this case, the equation for
1047:
4299:
Linear
Combinations and Span: Understanding linear combinations and spans of vectors
4645:
4534:
4484:
4377:
4270:
4037:
3484:
3137:. Similarly, we can speak of linear dependence or independence of an arbitrary set
405:); in this case one is probably referring to the expression, since every vector in
118:
106:
86:
3472:
are closed under conical combination but not affine or linear – hence one defines
4625:
4590:
4547:
4392:
3345:
2625:
2586:
536:
437:
38:
4654:
4397:
4178:
3440:
3299:
1401:
433:
413:
3952:{\displaystyle a_{1}\mathbf {v} _{1}b_{1}+\cdots +a_{n}\mathbf {v} _{n}b_{n}}
1941:{\displaystyle a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).}
4452:
1040:
1307:. To see this, suppose that 3 could be written as a linear combination of
4620:
3849:
425:
2548:
which is always false. Therefore, there is no way for this to work, and
1771:{\displaystyle (a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1}
3445:
3358:
2066:{\displaystyle a_{3}=1,\quad a_{2}+a_{3}=0,\quad a_{1}+a_{2}+a_{3}=-1.}
1193:{\displaystyle \cos t={\tfrac {1}{2}}\,e^{it}+{\tfrac {1}{2}}\,e^{-it}}
2276: − 1? If we try to make this vector a linear combination of
165:
linear combination of those vectors with those scalars as coefficients
4630:
397:, where nothing is specified (except that the vectors must belong to
374:
365:, with the coefficients unspecified (except that they must belong to
3188:, and the associated notions of sets closed under these operations.
424:
that the vectors are taken from (if one is mentioned) can still be
2094:, which comes out to −1. Finally, the last equation tells us that
4304:
2297:, then following the same process as before, we get the equation
2080:
can easily be solved. First, the first equation simply says that
81:
are constants). The concept of linear combinations is central to
4275:
3739:
algebraic operations in a vector space are linear combinations.
4308:
3866:. In that case, the most general linear combination looks like
2230:{\displaystyle x^{2}-1=-1-(x+1)+(x^{2}+x+1)=-p_{1}-p_{2}+p_{3}}
1955:
their corresponding coefficients are equal, so we can conclude
451:
2623:
linear combinations of these vectors. This set is called the
1620:{\displaystyle a_{1}(1)+a_{2}(x+1)+a_{3}(x^{2}+x+1)=x^{2}-1.}
93:, with some generalizations given at the end of the article.
3101:{\displaystyle \mathbf {0} =\sum _{i}c_{i}\mathbf {v} _{i}.}
412:
Note that by definition, a linear combination involves only
4222:
Lay, David C.; Lay, Steven R.; McDonald, Judi J. (2016).
2087:
is 1. Knowing that, we can solve the second equation for
3408:
3261:
1162:
1133:
3875:
3704:
3612:
3559:
3526:
3375:
3316:
3224:
3054:
2998:
2872:
2674:
2525:
2306:
2110:
1964:
1794:
1639:
1509:
1209:
1119:
649:
176:
4140:
4116:
1315:. This means that there would exist complex scalars
1101:, a square root of −1.) Some linear combinations of
30:"Superposition" redirects here. For other uses, see
4688:
4644:
4581:
4533:
4475:
4360:
3766:, then there may be a way to make sense of certain
3606:for instance corresponds to the linear combination
409:is certainly the value of some linear combination.
3951:
3719:
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3598:
3541:
3430:
3394:
3335:
3283:
3239:
3100:
3037:
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2825:
2537:
2501:
2229:
2065:
1940:
1770:
1619:
1283:
1192:
993:
615:To see that this is so, take an arbitrary vector (
285:
1630:Multiplying the polynomials out, this means
3045:), to saying a non-trivial combination is zero:
479:but its sources remain unclear because it lacks
1479: − 1. Picking arbitrary coefficients
4068:
1295:On the other hand, the constant function 3 is
1284:{\displaystyle 2\sin t=(-i)e^{it}+(i)e^{-it}.}
4320:
2272:On the other hand, what about the polynomial
8:
3774:. For example, we might be able to speak of
2817:
2723:
2619:). It is interesting to consider the set of
4248:(5th ed.). Wellesley Cambridge Press.
3841:scalar multiplication on the correct side.
3770:linear combinations, using the topology of
3747:for the operad of all linear combinations.
27:Sum of terms, each multiplied with a scalar
4327:
4313:
4305:
2992:This is equivalent, by subtracting these (
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3902:
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2002:
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978:
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841:
807:
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710:
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661:
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648:
510:Learn how and when to remove this message
416:many vectors (except as described in the
274:
269:
262:
243:
238:
231:
218:
213:
206:
193:
188:
181:
175:
4201:A (Terse) Introduction to Linear Algebra
3190:
3172:Affine, conical, and convex combinations
3148:is linearly independent and the span of
4049:
3516:, one can consider vector spaces to be
2847:Suppose that, for some sets of vectors
1454: − 1 a linear combination of
1404:with coefficients taken from the field
1368:, and clearly this cannot happen. See
4718:Comparison of linear algebra libraries
4056:
3542:{\displaystyle \mathbf {R} ^{\infty }}
4128:
4104:
4080:
1408:. Consider the vectors (polynomials)
389:(and the coefficients must belong to
7:
4092:
3844:A more complicated twist comes when
3512:More abstractly, in the language of
417:
401:and the coefficients must belong to
383:a linear combination of vectors in S
65:would be any expression of the form
4224:Linear Algebra and its Applications
1053:. Consider the vectors (functions)
393:). Finally, we may speak simply of
348:a linear combination of the vectors
4141:Katznelson & Katznelson (2008)
4117:Katznelson & Katznelson (2008)
3534:
3038:{\displaystyle c_{i}:=a_{i}-b_{i}}
25:
4199:; Katznelson, Yonatan R. (2008).
3599:{\displaystyle (2,3,-5,0,\dots )}
4731:
4730:
4708:Basic Linear Algebra Subprograms
4466:
3929:
3888:
3720:{\displaystyle \mathbf {R} ^{n}}
3707:
3672:
3654:
3636:
3618:
3529:
3240:{\displaystyle \mathbf {R} ^{n}}
3227:
3085:
3056:
2938:
2903:
2874:
2769:
2738:
2707:
2686:
974:
949:
924:
456:
270:
239:
214:
189:
113:. As usual, we call elements of
4606:Seven-dimensional cross product
3487:), generally the real numbers.
2017:
1984:
4246:Introduction to Linear Algebra
4069:Lay, Lay & McDonald (2016)
3593:
3560:
2717:
2681:
2489:
2480:
2425:
2386:
2377:
2351:
2182:
2157:
2151:
2139:
1932:
1923:
1892:
1853:
1844:
1818:
1781:and collecting like powers of
1746:
1694:
1688:
1659:
1653:
1640:
1595:
1570:
1554:
1542:
1526:
1520:
1259:
1253:
1234:
1225:
899:
881:
865:
847:
831:
813:
790:
765:
759:
734:
728:
703:
693:
654:
32:superposition (disambiguation)
1:
4205:American Mathematical Society
3836:instead of vector spaces. If
1091:base of the natural logarithm
4448:Eigenvalues and eigenvectors
3197:Restrictions on coefficients
2595:, an arbitrary vector space
448:Examples and counterexamples
3395:{\displaystyle a_{i}\geq 0}
3336:{\displaystyle a_{i}\geq 0}
591:is a linear combination of
539:, and let the vector space
18:Infinite linear combination
4773:
3505:
3111:If that is possible, then
2840:
2584:
2078:system of linear equations
1951:Two polynomials are equal
420:section. However, the set
29:
4726:
4464:
4342:
4226:(5th ed.). Pearson.
4179:10.1007/978-3-319-11080-6
4167:Linear Algebra Done Right
3466:probability distributions
3431:{\textstyle \sum a_{i}=1}
3284:{\textstyle \sum a_{i}=1}
4131:pp. 32-33, §§ 2.17, 2.19
3764:topological vector space
2653:}. We write the span of
2591:Take an arbitrary field
2556:a linear combination of
2248:a linear combination of
1392:, or any field, and let
1346:= π gives the equations
1299:a linear combination of
1093:, about 2.71828..., and
465:This section includes a
3476:as the linear closure.
3453:Because these are more
550:. Consider the vectors
494:more precise citations.
4433:Row and column vectors
4107:pp. 29-30, §§ 2.5, 2.8
3953:
3721:
3692:
3600:
3543:
3432:
3396:
3337:
3285:
3241:
3133:; otherwise, they are
3102:
3039:
2983:
2827:
2633:) of the vectors, say
2539:
2503:
2231:
2067:
1942:
1772:
1621:
1285:
1194:
995:
418:§ Generalizations
338:In a given situation,
287:
163:are scalars, then the
4438:Row and column spaces
4383:Scalar multiplication
4271:"Linear Combinations"
3954:
3722:
3693:
3601:
3544:
3433:
3397:
3338:
3286:
3242:
3103:
3040:
2984:
2828:
2540:
2504:
2232:
2068:
1943:
1773:
1622:
1334:for all real numbers
1286:
1195:
996:
288:
122:and call elements of
4573:Gram–Schmidt process
4525:Gaussian elimination
4095:Linear combinations.
3873:
3702:
3610:
3557:
3524:
3406:
3373:
3314:
3259:
3222:
3135:linearly independent
3052:
2996:
2870:
2672:
2523:
2304:
2108:
1962:
1792:
1637:
1507:
1207:
1117:
1037:continuous functions
647:
395:a linear combination
174:
4703:Numerical stability
4583:Multilinear algebra
4558:Inner product space
4408:Linear independence
4197:Katznelson, Yitzhak
3307:Conical combination
3194:Type of combination
3182:conical combination
2843:Linear independence
2837:Linear independence
2538:{\displaystyle 0=1}
53:constructed from a
4413:Linear combination
4301:, khanacademy.org.
4163:Axler, Sheldon Jay
3949:
3717:
3688:
3596:
3539:
3428:
3392:
3366:Convex combination
3333:
3281:
3252:Affine combination
3237:
3208:Linear combination
3186:convex combination
3178:affine combination
3130:linearly dependent
3098:
3072:
3035:
2979:
2925:
2890:
2823:
2552: − 1 is
2535:
2499:
2497:
2227:
2063:
1938:
1768:
1617:
1450:Is the polynomial
1281:
1190:
1171:
1142:
991:
989:
467:list of references
381:, we may speak of
283:
43:linear combination
4744:
4743:
4611:Geometric algebra
4568:Kronecker product
4403:Linear projection
4388:Vector projection
4279:. 27 October 2015
4255:978-0-9802327-7-6
4233:978-0-321-98238-4
4214:978-0-8218-4419-9
4188:978-3-319-11079-0
3470:positive measures
3451:
3450:
3063:
2951:
2950: where
2916:
2881:
1415: := 1,
1170:
1141:
523:Euclidean vectors
520:
519:
512:
316:linear dependence
16:(Redirected from
4764:
4734:
4733:
4616:Exterior algebra
4553:Hadamard product
4470:
4458:Linear equations
4329:
4322:
4315:
4306:
4288:
4286:
4284:
4259:
4237:
4218:
4192:
4169:(3rd ed.).
4144:
4138:
4132:
4126:
4120:
4114:
4108:
4102:
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3938:
3937:
3932:
3926:
3925:
3907:
3906:
3897:
3896:
3891:
3885:
3884:
3852:over two rings,
3827:commutative ring
3726:
3724:
3723:
3718:
3716:
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3520:over the operad
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2899:
2889:
2877:
2832:
2830:
2829:
2824:
2810:
2809:
2791:
2790:
2778:
2777:
2772:
2766:
2765:
2747:
2746:
2741:
2735:
2734:
2716:
2715:
2710:
2695:
2694:
2689:
2544:
2542:
2541:
2536:
2508:
2506:
2505:
2500:
2498:
2467:
2466:
2451:
2450:
2436:
2424:
2423:
2411:
2410:
2398:
2397:
2376:
2375:
2363:
2362:
2347:
2346:
2337:
2336:
2324:
2323:
2310:
2236:
2234:
2233:
2228:
2226:
2225:
2213:
2212:
2200:
2199:
2169:
2168:
2120:
2119:
2072:
2070:
2069:
2064:
2053:
2052:
2040:
2039:
2027:
2026:
2007:
2006:
1994:
1993:
1974:
1973:
1947:
1945:
1944:
1939:
1910:
1909:
1891:
1890:
1878:
1877:
1865:
1864:
1843:
1842:
1830:
1829:
1814:
1813:
1804:
1803:
1777:
1775:
1774:
1769:
1761:
1760:
1745:
1744:
1729:
1728:
1716:
1715:
1706:
1705:
1687:
1686:
1671:
1670:
1652:
1651:
1626:
1624:
1623:
1618:
1610:
1609:
1582:
1581:
1569:
1568:
1541:
1540:
1519:
1518:
1446:
1428:
1370:Euler's identity
1367:
1356:
1333:
1290:
1288:
1287:
1282:
1277:
1276:
1249:
1248:
1199:
1197:
1196:
1191:
1189:
1188:
1172:
1163:
1157:
1156:
1143:
1134:
1000:
998:
997:
992:
990:
983:
982:
977:
971:
970:
958:
957:
952:
946:
945:
933:
932:
927:
921:
920:
905:
880:
879:
846:
845:
812:
811:
796:
789:
788:
752:
751:
715:
714:
692:
691:
679:
678:
666:
665:
579:
569:
559:
515:
508:
504:
501:
495:
490:this section by
481:inline citations
460:
459:
452:
292:
290:
289:
284:
279:
278:
273:
267:
266:
248:
247:
242:
236:
235:
223:
222:
217:
211:
210:
198:
197:
192:
186:
185:
147:are vectors and
21:
4772:
4771:
4767:
4766:
4765:
4763:
4762:
4761:
4747:
4746:
4745:
4740:
4722:
4684:
4640:
4577:
4529:
4471:
4462:
4428:Change of basis
4418:Multilinear map
4356:
4338:
4333:
4295:
4282:
4280:
4269:
4266:
4256:
4242:Strang, Gilbert
4240:
4234:
4221:
4215:
4195:
4189:
4161:
4158:
4153:
4148:
4147:
4139:
4135:
4127:
4123:
4115:
4111:
4103:
4099:
4091:
4087:
4079:
4075:
4067:
4063:
4055:
4051:
4046:
4034:
4023:
4014:
4007:
4000:
3991:
3984:
3977:
3968:
3939:
3927:
3917:
3898:
3886:
3876:
3871:
3870:
3865:
3858:
3812:
3806:
3799:
3793:
3786:
3780:
3756:
3754:Generalizations
3729:convex polytope
3705:
3700:
3699:
3670:
3652:
3634:
3616:
3608:
3607:
3555:
3554:
3527:
3522:
3521:
3510:
3504:
3474:signed measures
3459:generalizations
3412:
3404:
3403:
3376:
3371:
3370:
3317:
3312:
3311:
3294:Affine subspace
3265:
3257:
3256:
3225:
3220:
3219:
3215:Vector subspace
3211:no restrictions
3174:
3126:
3117:
3083:
3073:
3050:
3049:
3025:
3012:
2999:
2994:
2993:
2966:
2953:
2936:
2926:
2901:
2891:
2868:
2867:
2862:
2853:
2845:
2839:
2801:
2782:
2767:
2757:
2736:
2726:
2705:
2684:
2670:
2669:
2652:
2643:
2615:be vectors (in
2614:
2605:
2589:
2583:
2581:The linear span
2576:
2569:
2562:
2521:
2520:
2496:
2495:
2458:
2442:
2437:
2429:
2428:
2415:
2402:
2389:
2367:
2354:
2338:
2328:
2315:
2302:
2301:
2296:
2289:
2282:
2268:
2261:
2254:
2244: − 1
2217:
2204:
2191:
2160:
2111:
2106:
2105:
2100:
2093:
2086:
2044:
2031:
2018:
1998:
1985:
1965:
1960:
1959:
1901:
1882:
1869:
1856:
1834:
1821:
1805:
1795:
1790:
1789:
1752:
1736:
1720:
1707:
1697:
1678:
1662:
1643:
1635:
1634:
1601:
1573:
1560:
1532:
1510:
1505:
1504:
1500:, we want
1499:
1492:
1485:
1474:
1467:
1460:
1436:
1430:
1422:
1416:
1414:
1378:
1358:
1347:
1324:
1291:
1262:
1237:
1205:
1204:
1200:
1174:
1145:
1115:
1114:
1030:
1019:complex numbers
1007:
988:
987:
972:
962:
947:
937:
922:
912:
903:
902:
871:
837:
803:
794:
793:
780:
743:
706:
696:
683:
670:
657:
645:
644:
635:
628:
621:
611:
604:
597:
577:
571:
567:
561:
557:
551:
545:Euclidean space
525:
516:
505:
499:
496:
485:
471:related reading
461:
457:
450:
364:
355:
334:
312:
303:
268:
258:
237:
227:
212:
202:
187:
177:
172:
171:
162:
153:
146:
137:
109:over the field
99:
35:
28:
23:
22:
15:
12:
11:
5:
4770:
4768:
4760:
4759:
4757:Linear algebra
4749:
4748:
4742:
4741:
4739:
4738:
4727:
4724:
4723:
4721:
4720:
4715:
4710:
4705:
4700:
4698:Floating-point
4694:
4692:
4686:
4685:
4683:
4682:
4680:Tensor product
4677:
4672:
4667:
4665:Function space
4662:
4657:
4651:
4649:
4642:
4641:
4639:
4638:
4633:
4628:
4623:
4618:
4613:
4608:
4603:
4601:Triple product
4598:
4593:
4587:
4585:
4579:
4578:
4576:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4539:
4537:
4531:
4530:
4528:
4527:
4522:
4517:
4515:Transformation
4512:
4507:
4505:Multiplication
4502:
4497:
4492:
4487:
4481:
4479:
4473:
4472:
4465:
4463:
4461:
4460:
4455:
4450:
4445:
4440:
4435:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4370:
4364:
4362:
4361:Basic concepts
4358:
4357:
4355:
4354:
4349:
4343:
4340:
4339:
4336:Linear algebra
4334:
4332:
4331:
4324:
4317:
4309:
4303:
4302:
4294:
4293:External links
4291:
4290:
4289:
4265:
4262:
4261:
4260:
4254:
4238:
4232:
4219:
4213:
4193:
4187:
4157:
4154:
4152:
4149:
4146:
4145:
4143:p. 14, § 1.3.2
4133:
4121:
4109:
4097:
4085:
4073:
4061:
4048:
4047:
4045:
4042:
4041:
4040:
4033:
4030:
4019:
4012:
4005:
3996:
3989:
3982:
3973:
3966:
3960:
3959:
3946:
3942:
3936:
3931:
3924:
3920:
3916:
3913:
3910:
3905:
3901:
3895:
3890:
3883:
3879:
3863:
3856:
3810:
3804:
3797:
3791:
3784:
3778:
3755:
3752:
3745:generating set
3714:
3709:
3687:
3684:
3679:
3674:
3669:
3666:
3661:
3656:
3651:
3648:
3643:
3638:
3633:
3630:
3625:
3620:
3615:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3549:(the infinite
3536:
3531:
3506:Main article:
3503:
3500:
3449:
3448:
3443:
3438:
3427:
3424:
3419:
3415:
3411:
3391:
3388:
3383:
3379:
3368:
3362:
3361:
3348:
3343:
3332:
3329:
3324:
3320:
3309:
3303:
3302:
3296:
3291:
3280:
3277:
3272:
3268:
3264:
3254:
3248:
3247:
3234:
3229:
3217:
3212:
3209:
3205:
3204:
3201:
3198:
3195:
3173:
3170:
3122:
3115:
3109:
3108:
3097:
3092:
3087:
3080:
3076:
3070:
3066:
3062:
3058:
3032:
3028:
3024:
3019:
3015:
3011:
3006:
3002:
2990:
2989:
2978:
2973:
2969:
2965:
2960:
2956:
2945:
2940:
2933:
2929:
2923:
2919:
2915:
2910:
2905:
2898:
2894:
2888:
2884:
2880:
2876:
2858:
2851:
2841:Main article:
2838:
2835:
2834:
2833:
2822:
2819:
2816:
2813:
2808:
2804:
2800:
2797:
2794:
2789:
2785:
2781:
2776:
2771:
2764:
2760:
2756:
2753:
2750:
2745:
2740:
2733:
2729:
2725:
2722:
2719:
2714:
2709:
2704:
2701:
2698:
2693:
2688:
2683:
2680:
2677:
2648:
2641:
2610:
2603:
2585:Main article:
2582:
2579:
2574:
2567:
2560:
2546:
2545:
2534:
2531:
2528:
2510:
2509:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2465:
2461:
2457:
2454:
2449:
2445:
2441:
2438:
2434:
2431:
2430:
2427:
2422:
2418:
2414:
2409:
2405:
2401:
2396:
2392:
2388:
2385:
2382:
2379:
2374:
2370:
2366:
2361:
2357:
2353:
2350:
2345:
2341:
2335:
2331:
2327:
2322:
2318:
2314:
2311:
2309:
2294:
2287:
2280:
2266:
2259:
2252:
2238:
2237:
2224:
2220:
2216:
2211:
2207:
2203:
2198:
2194:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2167:
2163:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2118:
2114:
2098:
2091:
2084:
2074:
2073:
2062:
2059:
2056:
2051:
2047:
2043:
2038:
2034:
2030:
2025:
2021:
2016:
2013:
2010:
2005:
2001:
1997:
1992:
1988:
1983:
1980:
1977:
1972:
1968:
1953:if and only if
1949:
1948:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1908:
1904:
1900:
1897:
1894:
1889:
1885:
1881:
1876:
1872:
1868:
1863:
1859:
1855:
1852:
1849:
1846:
1841:
1837:
1833:
1828:
1824:
1820:
1817:
1812:
1808:
1802:
1798:
1785:, we get
1779:
1778:
1767:
1764:
1759:
1755:
1751:
1748:
1743:
1739:
1735:
1732:
1727:
1723:
1719:
1714:
1710:
1704:
1700:
1696:
1693:
1690:
1685:
1681:
1677:
1674:
1669:
1665:
1661:
1658:
1655:
1650:
1646:
1642:
1628:
1627:
1616:
1613:
1608:
1604:
1600:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1572:
1567:
1563:
1559:
1556:
1553:
1550:
1547:
1544:
1539:
1535:
1531:
1528:
1525:
1522:
1517:
1513:
1497:
1490:
1483:
1472:
1465:
1458:
1434:
1420:
1412:
1377:
1374:
1293:
1292:
1280:
1275:
1272:
1269:
1265:
1261:
1258:
1255:
1252:
1247:
1244:
1240:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1203:
1201:
1187:
1184:
1181:
1177:
1169:
1166:
1160:
1155:
1152:
1148:
1140:
1137:
1131:
1128:
1125:
1122:
1113:
1099:imaginary unit
1026:
1006:
1003:
1002:
1001:
986:
981:
976:
969:
965:
961:
956:
951:
944:
940:
936:
931:
926:
919:
915:
911:
908:
906:
904:
901:
898:
895:
892:
889:
886:
883:
878:
874:
870:
867:
864:
861:
858:
855:
852:
849:
844:
840:
836:
833:
830:
827:
824:
821:
818:
815:
810:
806:
802:
799:
797:
795:
792:
787:
783:
779:
776:
773:
770:
767:
764:
761:
758:
755:
750:
746:
742:
739:
736:
733:
730:
727:
724:
721:
718:
713:
709:
705:
702:
699:
697:
695:
690:
686:
682:
677:
673:
669:
664:
660:
656:
653:
652:
633:
626:
619:
609:
602:
595:
575:
565:
555:
527:Let the field
524:
521:
518:
517:
475:external links
464:
462:
455:
449:
446:
360:
353:
330:
308:
301:
294:
293:
282:
277:
272:
265:
261:
257:
254:
251:
246:
241:
234:
230:
226:
221:
216:
209:
205:
201:
196:
191:
184:
180:
158:
151:
142:
135:
98:
95:
83:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4769:
4758:
4755:
4754:
4752:
4737:
4729:
4728:
4725:
4719:
4716:
4714:
4713:Sparse matrix
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4695:
4693:
4691:
4687:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4652:
4650:
4648:constructions
4647:
4643:
4637:
4636:Outermorphism
4634:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4596:Cross product
4594:
4592:
4589:
4588:
4586:
4584:
4580:
4574:
4571:
4569:
4566:
4564:
4563:Outer product
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4544:
4543:Orthogonality
4541:
4540:
4538:
4536:
4532:
4526:
4523:
4521:
4520:Cramer's rule
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4490:Decomposition
4488:
4486:
4483:
4482:
4480:
4478:
4474:
4469:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4365:
4363:
4359:
4353:
4350:
4348:
4345:
4344:
4341:
4337:
4330:
4325:
4323:
4318:
4316:
4311:
4310:
4307:
4300:
4297:
4296:
4292:
4278:
4277:
4272:
4268:
4267:
4263:
4257:
4251:
4247:
4243:
4239:
4235:
4229:
4225:
4220:
4216:
4210:
4206:
4202:
4198:
4194:
4190:
4184:
4180:
4176:
4172:
4168:
4164:
4160:
4159:
4155:
4150:
4142:
4137:
4134:
4130:
4125:
4122:
4119:p. 9, § 1.2.3
4118:
4113:
4110:
4106:
4101:
4098:
4094:
4089:
4086:
4082:
4077:
4074:
4070:
4065:
4062:
4058:
4057:Strang (2016)
4053:
4050:
4043:
4039:
4036:
4035:
4031:
4029:
4027:
4022:
4018:
4011:
4004:
3999:
3995:
3988:
3981:
3976:
3972:
3965:
3944:
3940:
3934:
3922:
3918:
3914:
3911:
3908:
3903:
3899:
3893:
3881:
3877:
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3842:
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3809:
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3777:
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3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3552:
3519:
3515:
3514:operad theory
3509:
3508:Operad theory
3502:Operad theory
3501:
3499:
3495:
3493:
3488:
3486:
3482:
3481:ordered field
3477:
3475:
3471:
3467:
3462:
3460:
3456:
3447:
3444:
3442:
3439:
3425:
3422:
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3409:
3389:
3386:
3381:
3377:
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3364:
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3356:
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3330:
3327:
3322:
3318:
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3308:
3305:
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3301:
3297:
3295:
3292:
3278:
3275:
3270:
3266:
3262:
3255:
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3216:
3213:
3210:
3207:
3206:
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3199:
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3125:
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3017:
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3004:
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2976:
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2963:
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2917:
2913:
2908:
2896:
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2866:
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2647:
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2598:
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2222:
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2209:
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2079:
2060:
2057:
2054:
2049:
2045:
2041:
2036:
2032:
2028:
2023:
2019:
2014:
2011:
2008:
2003:
1999:
1995:
1990:
1986:
1981:
1978:
1975:
1970:
1966:
1958:
1957:
1956:
1954:
1935:
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1496:
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1471:
1464:
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1433:
1426:
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1407:
1403:
1399:
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1391:
1387:
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1375:
1373:
1371:
1365:
1361:
1354:
1350:
1345:
1341:
1337:
1331:
1327:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1278:
1273:
1270:
1267:
1263:
1256:
1250:
1245:
1242:
1238:
1231:
1228:
1222:
1219:
1216:
1213:
1210:
1202:
1185:
1182:
1179:
1175:
1167:
1164:
1158:
1153:
1150:
1146:
1138:
1135:
1129:
1126:
1123:
1120:
1112:
1111:
1110:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1049:
1048:complex plane
1045:
1042:
1038:
1034:
1029:
1024:
1020:
1016:
1012:
1004:
984:
979:
967:
963:
959:
954:
942:
938:
934:
929:
917:
913:
909:
907:
896:
893:
890:
887:
884:
876:
872:
868:
862:
859:
856:
853:
850:
842:
838:
834:
828:
825:
822:
819:
816:
808:
804:
800:
798:
785:
781:
777:
774:
771:
768:
762:
756:
753:
748:
744:
740:
737:
731:
725:
722:
719:
716:
711:
707:
700:
698:
688:
684:
680:
675:
671:
667:
662:
658:
643:
642:
641:
640:, and write:
639:
632:
625:
618:
613:
608:
601:
594:
590:
586:
583:
574:
564:
554:
549:
546:
542:
538:
534:
530:
522:
514:
511:
503:
493:
489:
483:
482:
476:
472:
468:
463:
454:
453:
447:
445:
443:
439:
435:
431:
427:
423:
419:
415:
410:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
363:
359:
352:
349:
345:
341:
336:
333:
329:
325:
321:
317:
311:
307:
300:
280:
275:
263:
259:
255:
252:
249:
244:
232:
228:
224:
219:
207:
203:
199:
194:
182:
178:
170:
169:
168:
166:
161:
157:
150:
145:
141:
134:
130:
129:
125:
121:
120:
116:
112:
108:
104:
96:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
47:superposition
44:
40:
33:
19:
4646:Vector space
4412:
4378:Vector space
4281:. Retrieved
4274:
4245:
4223:
4200:
4166:
4136:
4129:Axler (2015)
4124:
4112:
4105:Axler (2015)
4100:
4088:
4083:p. 28, § 2.3
4081:Axler (2015)
4076:
4071:p. 28, ch. 1
4064:
4052:
4038:Weighted sum
4025:
4020:
4016:
4009:
4002:
3997:
3993:
3986:
3979:
3974:
3970:
3963:
3961:
3860:
3853:
3845:
3843:
3837:
3830:
3822:
3820:
3814:
3807:
3801:
3794:
3788:
3781:
3775:
3771:
3767:
3759:
3757:
3749:
3741:
3737:all possible
3736:
3733:
3511:
3496:
3489:
3485:ordered ring
3478:
3463:
3458:
3454:
3452:
3203:Model space
3175:
3165:
3157:
3153:
3149:
3145:
3143:
3141:of vectors.
3138:
3134:
3128:
3123:
3119:
3112:
3110:
2991:
2859:
2855:
2848:
2846:
2662:
2658:
2654:
2649:
2645:
2638:
2634:
2630:
2624:
2620:
2616:
2611:
2607:
2600:
2596:
2592:
2590:
2571:
2564:
2557:
2553:
2549:
2547:
2513:
2511:
2291:
2284:
2277:
2273:
2271:
2263:
2256:
2249:
2245:
2241:
2239:
2095:
2088:
2081:
2075:
1950:
1782:
1780:
1629:
1494:
1487:
1480:
1476:
1469:
1462:
1455:
1451:
1449:
1442:
1438:
1431:
1424:
1417:
1409:
1405:
1397:
1393:
1389:
1385:
1381:
1379:
1363:
1359:
1352:
1348:
1343:
1339:
1335:
1329:
1325:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1294:
1106:
1102:
1094:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1043:
1032:
1027:
1025:be the set C
1022:
1014:
1010:
1008:
637:
630:
623:
616:
614:
606:
599:
592:
588:
581:
572:
562:
552:
547:
540:
537:real numbers
532:
528:
526:
506:
497:
486:Please help
478:
441:
429:
421:
411:
406:
402:
398:
394:
390:
386:
382:
378:
370:
366:
361:
357:
350:
347:
343:
339:
337:
331:
327:
323:
319:
309:
305:
298:
295:
164:
159:
155:
148:
143:
139:
132:
126:
123:
117:
114:
110:
107:vector space
102:
100:
87:vector space
78:
74:
70:
66:
62:
58:
46:
42:
36:
4626:Multivector
4591:Determinant
4548:Dot product
4393:Linear span
4093:nLab (2015)
4059:p. 3, § 1.1
3346:Convex cone
3200:Name of set
3127:are called
2626:linear span
2587:Linear span
2570:, and
2262:, and
1402:polynomials
1396:be the set
1376:Polynomials
1109: are:
1061:defined by
1013:be the set
605:, and
531:be the set
500:August 2013
492:introducing
438:zero vector
318:: a family
39:mathematics
4660:Direct sum
4495:Invertible
4398:Linear map
4151:References
4024:belong to
4001:belong to
3978:belong to
3815:convergent
3551:direct sum
3455:restricted
3441:Convex set
3300:hyperplane
2599:, and let
1338:. Setting
1323:such that
1081:) :=
1069:) :=
1021:, and let
432:cannot be
369:). Or, if
97:Definition
51:expression
4690:Numerical
4453:Transpose
4044:Citations
3912:⋯
3686:⋯
3647:−
3591:…
3576:−
3535:∞
3410:∑
3387:≥
3328:≥
3263:∑
3065:∑
3023:−
2964:≠
2918:∑
2883:∑
2812:∈
2796:…
2752:⋯
2700:…
2679:
2629:(or just
2516: is
2484:−
2202:−
2189:−
2137:−
2131:−
2122:−
2058:−
1927:−
1763:−
1612:−
1437: :=
1423: :=
1268:−
1229:−
1217:
1180:−
1124:
1085:. (Here,
1041:real line
1039:from the
1035:) of all
1005:Functions
578:= (0,0,1)
568:= (0,1,0)
558:= (1,0,0)
253:⋯
4751:Category
4736:Category
4675:Subspace
4670:Quotient
4621:Bivector
4535:Bilinear
4477:Matrices
4352:Glossary
4244:(2016).
4171:Springer
4165:(2015).
4156:Textbook
4032:See also
3850:bimodule
3800: +
3787: +
3768:infinite
3518:algebras
3351:Quadrant
2661:) or sp(
2657:as span(
1342:= 0 and
426:infinite
414:finitely
73:, where
4347:Outline
3834:modules
3446:Simplex
3359:orthant
3298:Affine
3156:, then
3152:equals
2644:, ...,
1400:of all
1097:is the
1089:is the
1046:to the
1017:of all
580:. Then
543:be the
488:improve
128:scalars
119:vectors
89:over a
4631:Tensor
4443:Kernel
4373:Vector
4368:Scalar
4283:16 Feb
4252:
4230:
4211:
4185:
4008:, and
3962:where
3355:octant
3184:, and
2290:, and
1493:, and
1468:, and
1429:, and
585:vector
375:subset
49:is an
4500:Minor
4485:Block
4423:Basis
3992:,...,
3969:,...,
3848:is a
3825:is a
3762:is a
3357:, or
3162:basis
3160:is a
3118:,...,
2854:,...,
2606:,...,
2076:This
636:) in
473:, or
373:is a
356:,...,
304:,...,
154:,...,
138:,...,
131:. If
105:be a
91:field
4655:Dual
4510:Rank
4285:2021
4276:nLab
4250:ISBN
4228:ISBN
4209:ISBN
4183:ISBN
3859:and
3492:cone
3483:(or
3402:and
3164:for
2676:span
2631:span
1380:Let
1366:= −3
1357:and
1319:and
1311:and
1303:and
1105:and
1073:and
1057:and
1009:Let
570:and
434:zero
342:and
101:Let
77:and
61:and
41:, a
4264:Web
4175:doi
4015:,…,
3821:If
3758:If
3144:If
2665:):
2637:= {
2621:all
2554:not
2240:so
1445:+ 1
1427:+ 1
1384:be
1355:= 3
1332:= 3
1297:not
1214:sin
1121:cos
587:in
582:any
535:of
440:in
377:of
167:is
55:set
45:or
37:In
4753::
4273:.
4207:.
4203:.
4181:.
4173:.
4028:.
3985:,
3353:,
3180:,
3168:.
3010::=
2721::=
2577:.
2563:,
2283:,
2269:.
2255:,
2246:is
2061:1.
1615:1.
1486:,
1461:,
1447:.
1441:+
1388:,
1372:.
1362:+
1351:+
1330:be
1328:+
1326:ae
612:.
598:,
560:,
477:,
469:,
444:.
71:by
69:+
67:ax
4328:e
4321:t
4314:v
4287:.
4258:.
4236:.
4217:.
4191:.
4177::
4026:V
4021:n
4017:v
4013:1
4010:v
4006:R
4003:K
3998:n
3994:b
3990:1
3987:b
3983:L
3980:K
3975:n
3971:a
3967:1
3964:a
3945:n
3941:b
3935:n
3930:v
3923:n
3919:a
3915:+
3909:+
3904:1
3900:b
3894:1
3889:v
3882:1
3878:a
3864:R
3861:K
3857:L
3854:K
3846:V
3838:K
3831:V
3823:K
3811:3
3808:v
3805:3
3802:a
3798:2
3795:v
3792:2
3789:a
3785:1
3782:v
3779:1
3776:a
3772:V
3760:V
3713:n
3708:R
3683:+
3678:4
3673:v
3668:0
3665:+
3660:3
3655:v
3650:5
3642:2
3637:v
3632:3
3629:+
3624:1
3619:v
3614:2
3594:)
3588:,
3585:0
3582:,
3579:5
3573:,
3570:3
3567:,
3564:2
3561:(
3530:R
3426:1
3423:=
3418:i
3414:a
3390:0
3382:i
3378:a
3331:0
3323:i
3319:a
3279:1
3276:=
3271:i
3267:a
3233:n
3228:R
3166:V
3158:S
3154:V
3150:S
3146:S
3139:S
3124:n
3120:v
3116:1
3113:v
3096:.
3091:i
3086:v
3079:i
3075:c
3069:i
3061:=
3057:0
3031:i
3027:b
3018:i
3014:a
3005:i
3001:c
2977:.
2972:i
2968:b
2959:i
2955:a
2944:i
2939:v
2932:i
2928:b
2922:i
2914:=
2909:i
2904:v
2897:i
2893:a
2887:i
2879:=
2875:v
2860:n
2856:v
2852:1
2849:v
2821:.
2818:}
2815:K
2807:n
2803:a
2799:,
2793:,
2788:1
2784:a
2780::
2775:n
2770:v
2763:n
2759:a
2755:+
2749:+
2744:1
2739:v
2732:1
2728:a
2724:{
2718:)
2713:n
2708:v
2703:,
2697:,
2692:1
2687:v
2682:(
2663:S
2659:S
2655:S
2650:n
2646:v
2642:1
2639:v
2635:S
2617:V
2612:n
2608:v
2604:1
2601:v
2597:V
2593:K
2575:3
2572:p
2568:2
2565:p
2561:1
2558:p
2550:x
2533:1
2530:=
2527:0
2514:x
2493:.
2490:)
2487:1
2481:(
2478:+
2475:x
2472:0
2469:+
2464:2
2460:x
2456:0
2453:+
2448:3
2444:x
2440:1
2433:=
2426:)
2421:3
2417:a
2413:+
2408:2
2404:a
2400:+
2395:1
2391:a
2387:(
2384:+
2381:x
2378:)
2373:3
2369:a
2365:+
2360:2
2356:a
2352:(
2349:+
2344:2
2340:x
2334:3
2330:a
2326:+
2321:3
2317:x
2313:0
2295:3
2292:p
2288:2
2285:p
2281:1
2278:p
2274:x
2267:3
2264:p
2260:2
2257:p
2253:1
2250:p
2242:x
2223:3
2219:p
2215:+
2210:2
2206:p
2197:1
2193:p
2186:=
2183:)
2180:1
2177:+
2174:x
2171:+
2166:2
2162:x
2158:(
2155:+
2152:)
2149:1
2146:+
2143:x
2140:(
2134:1
2128:=
2125:1
2117:2
2113:x
2099:1
2096:a
2092:2
2089:a
2085:3
2082:a
2055:=
2050:3
2046:a
2042:+
2037:2
2033:a
2029:+
2024:1
2020:a
2015:,
2012:0
2009:=
2004:3
2000:a
1996:+
1991:2
1987:a
1982:,
1979:1
1976:=
1971:3
1967:a
1936:.
1933:)
1930:1
1924:(
1921:+
1918:x
1915:0
1912:+
1907:2
1903:x
1899:1
1896:=
1893:)
1888:3
1884:a
1880:+
1875:2
1871:a
1867:+
1862:1
1858:a
1854:(
1851:+
1848:x
1845:)
1840:3
1836:a
1832:+
1827:2
1823:a
1819:(
1816:+
1811:2
1807:x
1801:3
1797:a
1783:x
1766:1
1758:2
1754:x
1750:=
1747:)
1742:3
1738:a
1734:+
1731:x
1726:3
1722:a
1718:+
1713:2
1709:x
1703:3
1699:a
1695:(
1692:+
1689:)
1684:2
1680:a
1676:+
1673:x
1668:2
1664:a
1660:(
1657:+
1654:)
1649:1
1645:a
1641:(
1607:2
1603:x
1599:=
1596:)
1593:1
1590:+
1587:x
1584:+
1579:2
1575:x
1571:(
1566:3
1562:a
1558:+
1555:)
1552:1
1549:+
1546:x
1543:(
1538:2
1534:a
1530:+
1527:)
1524:1
1521:(
1516:1
1512:a
1498:3
1495:a
1491:2
1488:a
1484:1
1481:a
1477:x
1473:3
1470:p
1466:2
1463:p
1459:1
1456:p
1452:x
1443:x
1439:x
1435:3
1432:p
1425:x
1421:2
1418:p
1413:1
1410:p
1406:K
1398:P
1394:V
1390:C
1386:R
1382:K
1364:b
1360:a
1353:b
1349:a
1344:t
1340:t
1336:t
1321:b
1317:a
1313:e
1309:e
1305:g
1301:f
1279:.
1274:t
1271:i
1264:e
1260:)
1257:i
1254:(
1251:+
1246:t
1243:i
1239:e
1235:)
1232:i
1226:(
1223:=
1220:t
1211:2
1186:t
1183:i
1176:e
1168:2
1165:1
1159:+
1154:t
1151:i
1147:e
1139:2
1136:1
1130:=
1127:t
1107:g
1103:f
1095:i
1087:e
1083:e
1079:t
1077:(
1075:g
1071:e
1067:t
1065:(
1063:f
1059:g
1055:f
1051:C
1044:R
1033:R
1031:(
1028:C
1023:V
1015:C
1011:K
985:.
980:3
975:e
968:3
964:a
960:+
955:2
950:e
943:2
939:a
935:+
930:1
925:e
918:1
914:a
910:=
900:)
897:1
894:,
891:0
888:,
885:0
882:(
877:3
873:a
869:+
866:)
863:0
860:,
857:1
854:,
851:0
848:(
843:2
839:a
835:+
832:)
829:0
826:,
823:0
820:,
817:1
814:(
809:1
805:a
801:=
791:)
786:3
782:a
778:,
775:0
772:,
769:0
766:(
763:+
760:)
757:0
754:,
749:2
745:a
741:,
738:0
735:(
732:+
729:)
726:0
723:,
720:0
717:,
712:1
708:a
704:(
701:=
694:)
689:3
685:a
681:,
676:2
672:a
668:,
663:1
659:a
655:(
638:R
634:3
631:a
629:,
627:2
624:a
622:,
620:1
617:a
610:3
607:e
603:2
600:e
596:1
593:e
589:R
576:3
573:e
566:2
563:e
556:1
553:e
548:R
541:V
533:R
529:K
513:)
507:(
502:)
498:(
484:.
442:V
430:n
422:S
407:V
403:K
399:V
391:K
387:S
379:V
371:S
367:K
362:n
358:v
354:1
351:v
344:V
340:K
332:i
328:v
324:F
320:F
310:n
306:v
302:1
299:v
281:.
276:n
271:v
264:n
260:a
256:+
250:+
245:3
240:v
233:3
229:a
225:+
220:2
215:v
208:2
204:a
200:+
195:1
190:v
183:1
179:a
160:n
156:a
152:1
149:a
144:n
140:v
136:1
133:v
124:K
115:V
111:K
103:V
79:b
75:a
63:y
59:x
34:.
20:)
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