95:
2481:). This fundamentally changes the definition of a transpose in the abstract context. It would make more sense to me if it were defined primarily in the metric-free context, and (if desired) related to the concept defined in the section at present when suitable bilinear forms are available. I suggest reverting this section to the earlier form, with the approach using bilinear forms omitted. Does anyone with more familiarity of the area know what the most generally accepted definition is? —
85:
64:
31:
22:
2797:
matrix angle to its complementary angle of pi/2 changing cos(θ) to sin(θ), allowing us to use simpler equations especially in inverse(A) = transpose(adjoint(A)) / det(A). An equation with sin(θ) and cos(θ) is vastly simpler than one with just cos(θ) to compute all other mathematical equations using tan(θ), tanh(θ), etc.
2493:
I would call what is described in that section of the article the adjoint rather than the transpose, although I'm not sure whether there is a universally accepted definition. It would make sense to me to define the transpose in a metric free setting and define the adjoint as a generalization. I'm a
585:
to the T power"? But I'm unfamiliar with this notation, and I certainly don't have the same objection to used with matrices. So, are there any experts who could weigh in with the most common usage in this area? (I'm not proposing to change the notation used in the earlier sections, just to keep the
1015:
Here's my confusion: If row vectors and column vectors are in distinct spaces (and they certainly, even in elementary linear algebra in that you can't just add a column to a row vector because they have different shapes), then taking the transpose of a vector isn't just some notational convenience,
2796:
The reason transpose of a matrix is used is to get the sin(θ) between two vectors in a matrix. The dot product gives the cos(θ) of two vectors, and if we want to get the sin(θ), we would have to do a cos inverse operation to get the angle or use sin^2 + cos^2 = 1. Transpose of a matrix rotates the
2218:
will not be diagonal (because off-diagonal terms are inner products of columns with different columns) in the case that the columns aren't orthogonal, and it will have something other than one on the diagonal in the case that any columns don't have norm of one (because the diagonals are the inner
3242:
This
Knowledge article can easily overwhelm a beginner. there is no point in giving all the unstructured & unstitched details. The article requires a section on motivation for transposes and the link to least squares and the four fundamental sub spaces. Can we work towards a well directed
1656:
does possess a preferred isomorphism with its dual, that is the transpose, if we represent n-vectors with columns and forms with rows. Fixing an isomorphism V→V* is the same as giving V a scalar product (check the equivalence), which is a tensor of type (0,2), that eats pairs of vectors and
1245:
of type (1,1) and all (co-) vectors are either of type (0,1) or (1,0). But that usage doesn't extend to problems involving things like type-(0,2) or type-(2,0) tensors since usual linear algebra doesn't allow for a row vector of row vectors. My hunch is that in this case, transpose is used a
1237:
Sure. I'm asking because I get the sense that there are some unwritten ruels going on. At one extreme is the purely-mechanical notion of tranpose that you describe, which I'm happy with. In that context, transpose is just used along with matrix operations to simplify the expression of some
1675:
Those are great answers! That clarifies some things that have been nagging me for a long time! I feel like It is particularly helpful to think that conventional matrix notation doesn't provide notation for a row of row vectors or the like. I will probably copy the above discussion to
3258:
Hello
Rupnagar. Using this Talk space to start, please explain the 4 fundamental subspaces. And what is the connection to least squares? Note that transpose refers to binary relations as well as linear transformations, so various details are required. As for motivation, the
2164:
Hopefully someone better versed in (multi-)linear algebra literature comes along and knows if there's a more general definition. If there isn't, or if it's still fully compatible, let's change the "if" to "iff" here and possibly in the "Orthogonal Matrix" page too.
1587:
753:
Here's my confusion: If row vectors and column vectors are in distinct spaces (and they certainly are in that you can't just add them), then taking the transpose of a vector isn't just some notational convenience, it is an application of a nontrivial function,
596:
Putting the T in front sounds like a worthy experiment. In general though, you have to realise that mathematics notation is never anywhere near as rigorous as people would like to think. A classic example is -1 designates both the inverse and the reciprocal.
1842:. For a complex number, I'm not sure what would generalize to "row vector" or "column vector"... I'm not sure what I'm asking, but I feel like there's a little more that could be said connecting the above great explanations to conjugate transpose. :-)
2518:
Thanks. It is a pity that the definitions seem to be a little variable (gauging from the few references I've browsed). I'll make a change along these lines in the next week or so, any comment from other editors being welcome. —
2823:
Devssh has taken an interest in this article, made the above contribution, thanked me for the correction, and today entered more unhelpful edits into the article. Only contributes here and has not taken up a user page.
2644:
225:
2598:
3022:
3155:
2530:
I've made some comprehensive changes to the section, criticism welcome. I also removed a misguided association of the transpose of a coordinate vector and the more abstract concept of a transpose from
546:
447:
1404:
1075:
813:
1160:
897:
3224:
2919:
331:
151:
1409:
987:
939:
I'm confused. It seems like much of linear algebra glosses over the meaning of transposition and simply uses it as a mechanism for manipulating numbers, for example, defining the norm of
729:
681:
I'm confused. It seems like much of linear algebra glosses over the meaning of transposition and simply uses it as a mechanism for manipulating numbers, for example, defining the norm of
3307:
2653:
1089:
Is it correct that there are are two things going on here: (1) using transpose for numerical convenience and (2) using rows versus columns to for indicateng co- versus contravariance?
826:
Is it correct that there are are two things going on here: (1) using transpose for numerical convenience and (2) using rows versus columns to for indicateng co- versus contravariance?
1625:
2391:
2359:
1840:
1905:
1298:
647:
2668:
I think this is a matter of notation: sometimes the star is used to denote complex conjugate. I've changed this in the interest of reducing ambiguity and misunderstanding. —
1949:
1927:
1876:
1331:
2216:
1784:
935:
that didn't get any responses there. Perhaps this is a better audience since it's a bit of an essoteric question for such an elementary topic; Here's the question again:
2323:
2303:
569:
470:
354:
3297:
1754:
1719:
2444:
2417:
3312:
2283:
2263:
2161:
are (one or both) diagonal or something. Not necessarily that one, but it's enough to make me suspect there's a more general definition some people might use.
35:
3322:
141:
3292:
2143:
is orthogonal if..." to "iff" but I was unsure if this was really a fundamental definition. The "Orthogonal Matrix" page does the same as this one.
995:
735:
1238:
operations. At the other extreme, rows and columns correspond to co- and contra-variant vectors, in which case transpose is completely non-trivial.
3302:
994:
In some linear-algebra topics, however, it appears that column and row vectors have different meanings (that appear to have something to do with
734:
In some linear-algebra topics, however, it appears that column and row vectors have different meanings (that appear to have something to do with
117:
3215:
is entirely appropriate. But the basic fact above, in its most common manifestation, ought to be mentioned much, much earlier in the article.
3317:
2603:
2471:, the abstract definition of a transpose is in principle independent of any bilinear form. This was stated in this way until changed by
2449:
186:
3244:
3228:
108:
69:
750:. I think the idea is that column vectors would be indexed with raised indices and row vectors with lowered indices with tensors.
2657:
2557:
1010:. I think the idea is that column vectors would be indexed with raised indices and row vectors with lowered indices with tensors.
2924:
3287:
3062:
2777:
1643:
Also note that there is no canonical isomorphism between V and V* if V is a plain real vector space of finite dimension : -->
2505:
2856:
Varga, A. (January 1996). "Computation of
Kronecker-like forms of a system pencil: applications, algorithms and software".
499:
400:
1241:
My hunch is that the co- and contravariance convention is useful for some limited cases in which all transformations are
1582:{\displaystyle \scriptstyle g_{ij}v^{i}v^{j}=(g_{ij}v^{i})v^{j}=(\delta _{ij}v^{i})v^{j}=v_{j}^{\top }v^{j}=v^{\top }v.}
1333:. Does that sound right, or am I jumping to conclusions? If this is right, it could do with some explanation somewhere.
183:
I was trying to understand what the derivative of a transposed matrix is with respect to that matrix? So something like
44:
1359:
1019:
757:
1099:
836:
2858:
Proceedings of Joint
Conference on Control Applications Intelligent Control and Computer Aided Control System Design
2883:
284:
3175:
with the definition from articles as one of the generalizations of the transposed matrix here in this article? --
1856:
A complex number (just as a real number) is a 1-D vector, so rows and columns are the same thing. The modulus on
2499:
1907:(ie. for n=1). From an algebraic point of view, complex conjugation is the unique (non-trivial) automorphism on
946:
688:
3168:
in the context of the generation of the Quasi-Kronecker form of a matrix pencil (see e.g., Varga's article).
2453:
2228:
1847:
1338:
1178:
912:
654:
172:
1955:. I'm not really sure what the importance and meaning is from a geometrical or analytic point of view... --
3260:
3248:
2762:
In the text, should "a matrix raised to the Ath power" read instead "a matrix raised to the T-th power"?
1596:
2735:
Clicking on this link calls for enrollment in a course. If someone enrolls and finds good information on
2773:
2170:
50:
2363:
2331:
1789:
94:
2105:
2765:
2710:
2649:
1881:
1253:
616:
602:
21:
2769:
2728:
2494:
little surprised that we don't already have an article on the adjoint (except for the special case
2166:
1684:
1632:
3180:
116:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
2224:
2098:
1932:
1910:
1859:
1843:
1334:
1174:
908:
650:
100:
1303:
84:
63:
2191:
1759:
3268:
2873:
The definition there is: "transposed with respect to the main antidiagonal". (Just search for
2833:
2810:
2748:
2669:
2536:
2520:
2495:
2482:
1688:
1353:
3176:
2847:
In some articles about numerical methods for control systems there is also the notion of the
2861:
2476:
2308:
2288:
2265:
is a terrible notation for anything because it looks like a zero. We should change this to
1662:
1589:
That "usual linear algebra doesn't allow for a row vector of row vectors" is the reason why
587:
552:
453:
337:
2805:. The business of sin(θ) being found this way is unclear and is given with no reference. —
2123:
In the "Special transpose matrices" section, the writing implies that an orthogonal matrix
1724:
1694:
670:
2802:
2706:
2457:
2422:
2395:
1960:
1680:
for postarity and will probably add explanation along these lines to appropriate articles.
1228:
1207:=1, then transposition is just what it is with any other matrix - a map from the space of
598:
371:
255:
3194:
For the transpose M of a real square matrix considered as a linear mapping M : R —: -->
2820:
2220:
1628:
3157:
without the transposition. Maybe that is just a typo. But, it caused me to search for
2268:
2248:
1191:
I guess it depends on how we define vectors. If we consider a vector as just being an
3281:
1952:
1687:
reminds me that I've also long been suspiscious of its "meaning" (and simply that of
1093:
830:
3264:
2829:
2806:
2744:
1983:
1644:
1, with no additional structure. What is of course canonical is the pairing VxV* →
1242:
375:
259:
229:
1658:
113:
2828:
is encouraged to communicate in this Talk space before contributing further. —
2865:
2532:
2057:
1998:
1956:
1224:
1078:
1003:
816:
743:
379:
263:
90:
3045:
2646:. I think this is also what user Moala pointed out below way back in 2005.
2468:
3204:
Yet in the article this fact is buried deep in the article, in the section
3029:
Teruel, Ginés R Pérez (2020). "Matrix operators and the Klein four group".
3195:
R, with the standard dot product ⟨v,w⟩, we have the standard fact that
2740:
2328:
Also, we need to be consistent for our transpose notation. Should it be
2088:
3243:
article? With a couple of volunteers I can take up this responsibility.
2639:{\displaystyle \mathbf {A} ^{\mathrm {T} }={\overline {\mathbf {A} }}}
2146:
It seems like a decent definition of an orthogonal matrix could be a
1590:
1247:
1092:
Isn't the conventional
Euclidean metric defined with a contravariant
829:
Isn't the conventional
Euclidean metric defined with a contravariant
2478:
Linear
Algebra Quick Study Guide for Smartphones and Mobile Devices
220:{\displaystyle {\frac {\mathrm {d} A\Theta }{\mathrm {d} \Theta }}}
2463:
Transpose of linear maps: why defined in terms of a bilinear form?
1786:
on a vector, using conjugate transpose as a mechanism to compute
1691:) for the same reasons. Could you comment on that? In some sense
3272:
3252:
3232:
3184:
2837:
2814:
2781:
2752:
2714:
2674:
2661:
2541:
2525:
2509:
2487:
2232:
2174:
2113:
1964:
1851:
1666:
1636:
1342:
1232:
1182:
916:
658:
606:
590:
232:
2188:
aren't orthogonal (or a column doesn't have norm of one). Then
1648:. Fixing a base on V is the same as fixing an isomorphism with
2005:
be the matrix representation of ƒ with respect to some bases {
15:
2019:}. I seem to recall, please do correct me if I'm wrong, that
2593:{\displaystyle \mathbf {A} ^{\mathrm {T} }=\mathbf {A} ^{*}}
1683:
I haven't worked much with complex tensors, but your use of
3017:{\displaystyle A^{P}=A({\rm {end}}:-1:1,{\rm {end}}:-1:1)'}
1593:
notation is used when a row vector of row vectors, such as
3150:{\displaystyle A^{P}=A({\rm {end}}:-1:1,{\rm {end}}:-1:1)}
2180:
I'm pretty sure you are right and that it is iff. Suppose
1162:? Doesn't that not involve any transposition in that both
899:? Doesn't that not involve any transposition in that both
2472:
2139:. Thus I was going to change the "if" in "...that is,
2367:
2335:
1600:
1413:
1363:
998:). In that context, the transpose of a column vector,
541:{\displaystyle f^{\mathrm {T} }(\phi )=\phi \circ f\,}
442:{\displaystyle f(\phi )^{\mathrm {T} }=\phi \circ f\,}
3065:
2927:
2886:
2606:
2560:
2425:
2398:
2366:
2334:
2311:
2291:
2271:
2251:
2194:
1935:
1913:
1884:
1862:
1792:
1762:
1727:
1697:
1652:, hence produces a special isomorphism V→V*, because
1599:
1412:
1362:
1306:
1256:
1102:
1022:
949:
839:
760:
691:
619:
555:
502:
456:
403:
340:
287:
189:
112:, a collaborative effort to improve the coverage of
1878:can be thought of as a special case of the norm on
3149:
3016:
2913:
2638:
2592:
2438:
2411:
2385:
2353:
2317:
2297:
2285:or something. Is the author trying to suggest an
2277:
2257:
2210:
1943:
1921:
1899:
1870:
1834:
1778:
1748:
1713:
1619:
1581:
1399:{\displaystyle \scriptstyle g_{ij}=\delta _{ij}\,}
1398:
1325:
1292:
1154:
1069:
981:
891:
807:
723:
641:
577:But even that seems ambiguous; is this "transpose
563:
540:
464:
441:
348:
325:
219:
171:
2739:, then it might be used. For now it is just
1109:
1103:
1081:, but that involves more structure than just
1002:, gives you a row vector -- a vector in the
956:
950:
932:
846:
840:
742:, gives you a row vector -- a vector in the
698:
692:
3219:because that is often how the transpose is
3047:L∞-Norm Computation for Descriptor Systems
2763:
982:{\displaystyle \|v\|={\sqrt {v^{\top }v}}}
724:{\displaystyle \|v\|={\sqrt {v^{\top }v}}}
58:
3117:
3116:
3086:
3085:
3070:
3064:
2979:
2978:
2948:
2947:
2932:
2926:
2899:
2895:
2894:
2885:
2724:The following external link was removed:
2626:
2624:
2614:
2613:
2608:
2605:
2584:
2579:
2568:
2567:
2562:
2559:
2430:
2424:
2403:
2397:
2372:
2368:
2365:
2340:
2336:
2333:
2310:
2305:? Because then they should just use the
2290:
2270:
2250:
2199:
2193:
1951:fixed. Such automorphisms are central to
1937:
1936:
1934:
1915:
1914:
1912:
1891:
1887:
1886:
1883:
1864:
1863:
1861:
1826:
1813:
1797:
1791:
1767:
1761:
1726:
1702:
1696:
1605:
1598:
1566:
1553:
1543:
1538:
1525:
1512:
1499:
1483:
1470:
1457:
1441:
1431:
1418:
1411:
1384:
1368:
1361:
1311:
1305:
1284:
1274:
1261:
1255:
1144:
1134:
1121:
1115:
1101:
1061:
1042:
1021:
968:
962:
948:
881:
871:
858:
852:
838:
799:
780:
759:
710:
704:
690:
624:
618:
554:
508:
507:
501:
455:
418:
417:
402:
339:
291:
289:
286:
206:
193:
190:
188:
2550:Error in definition of Hermitian matrix?
996:covariance and contravariance of vectors
736:covariance and contravariance of vectors
3298:Knowledge vital articles in Mathematics
2241:
2184:is not orthogonal. Then two columns of
1614:
1393:
536:
437:
321:
60:
19:
3027:This is also the way it is defined in
2242:Notation in 'Transpose of Linear Maps'
2219:products of columns with themselves).
179:Differentiation of transposed matrices
3313:C-Class vital articles in Mathematics
3225:2601:200:C000:1A0:291B:4FAF:4C47:67FE
1620:{\displaystyle \scriptstyle g_{ij}\,}
738:). The transpose of a column vector,
7:
3171:Wouldn't it be great to mention the
490:)), and it would be better to write
478:Now, I think that's confusing (it's
106:This article is within the scope of
3208:, only in the greatest generality.
2854:An example for such an article is:
1300:to be represented with matrices as
922:As asked on the Math Reference Desk
49:It is of interest to the following
3124:
3121:
3118:
3093:
3090:
3087:
2986:
2983:
2980:
2955:
2952:
2949:
2654:2601:9:2C80:464:809A:FAF:9F56:4439
2615:
2569:
2386:{\displaystyle \textstyle {^{T}}v}
2354:{\displaystyle \textstyle {^{t}}v}
2200:
1835:{\displaystyle c^{*}c=a^{2}+b^{2}}
1768:
1567:
1544:
1312:
1043:
969:
781:
711:
625:
586:notation in this one section.) --
509:
419:
362:An anonymous user changed that to
211:
207:
201:
194:
14:
3323:Mid-priority mathematics articles
126:Knowledge:WikiProject Mathematics
3293:Knowledge level-5 vital articles
3031:Palestine Journal of Mathematics
2627:
2609:
2580:
2563:
2475:(which may have been taken from
2083:is the evaluation of the vector
1900:{\displaystyle \mathbb {C} ^{n}}
1293:{\displaystyle g_{ij}v^{i}v^{j}}
642:{\displaystyle f^{\top }(\phi )}
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
2705:Should that be ′ or ʹ? Source?
246:section, the article had read:
146:This article has been rated as
3303:C-Class level-5 vital articles
3273:04:02, 19 September 2022 (UTC)
3253:05:46, 18 September 2022 (UTC)
3233:18:18, 24 September 2021 (UTC)
3144:
3082:
3007:
2944:
2801:Perhaps the contributor means
2068:respectively. In this setting
1518:
1492:
1476:
1450:
1054:
1032:
1026:
792:
770:
764:
673:09:23, 20 December 2005 (UTC)
636:
630:
521:
515:
414:
407:
306:
300:
1:
3044:Voigt, Matthias (June 2010).
2815:22:39, 22 February 2019 (UTC)
2114:23:26, 23 November 2009 (UTC)
1965:19:41, 23 November 2009 (UTC)
1852:19:19, 23 November 2009 (UTC)
1667:18:46, 23 November 2009 (UTC)
1637:16:53, 23 November 2009 (UTC)
1343:15:13, 23 November 2009 (UTC)
1233:14:38, 23 November 2009 (UTC)
1183:14:16, 23 November 2009 (UTC)
917:05:00, 11 November 2009 (UTC)
233:13:26, 20 November 2006 (UTC)
120:and see a list of open tasks.
3318:C-Class mathematics articles
2782:21:42, 28 January 2019 (UTC)
2758:Matrix to the Ath power ...?
2715:08:18, 9 November 2017 (UTC)
2631:
2458:19:16, 10 October 2012 (UTC)
1944:{\displaystyle \mathbb {R} }
1922:{\displaystyle \mathbb {C} }
1871:{\displaystyle \mathbb {C} }
3185:08:14, 8 January 2021 (UTC)
2921:I get the pertransposed as
2792:The following was removed:
1326:{\displaystyle v^{\top }gv}
3339:
3201:for all vectors v, w ∊ R.
2211:{\displaystyle G^{\top }G}
1779:{\displaystyle v^{\top }v}
1250:to allow expressions like
933:question on Talk:Transpose
591:21:14, 8 August 2005 (UTC)
382:W* and V*, we define the
266:W* and V*, we define the
2866:10.1109/CACSD.1996.555201
2838:01:01, 7 March 2019 (UTC)
2753:22:59, 24 July 2018 (UTC)
2600:but I think it should be
2233:20:05, 16 June 2011 (UTC)
2175:19:20, 16 June 2011 (UTC)
1756:is the same operation as
1215:matrices to the space of
175:08:39, 11 Mar 2004 (UTC)
145:
78:
57:
2675:04:41, 19 May 2014 (UTC)
2662:03:42, 19 May 2014 (UTC)
2542:02:00, 5 June 2013 (UTC)
2526:11:07, 2 June 2013 (UTC)
2510:14:59, 1 June 2013 (UTC)
2488:14:02, 1 June 2013 (UTC)
2469:Transpose of linear maps
659:12:40, 11 May 2010 (UTC)
607:04:52, 11 May 2010 (UTC)
244:Transpose of linear maps
238:Transpose of linear maps
227:where both are matrices.
152:project's priority scale
2729:How to create Transpose
2318:{\displaystyle \omega }
2298:{\displaystyle \omega }
677:Transpose on tensors...
564:{\displaystyle \ \phi }
482:that's transposed, not
465:{\displaystyle \ \phi }
349:{\displaystyle \ \phi }
109:WikiProject Mathematics
3288:C-Class vital articles
3261:relation (mathematics)
3151:
3024:with Octave notation.
3018:
2915:
2877:within that article.)
2681:Which Prime character?
2640:
2594:
2440:
2413:
2387:
2355:
2319:
2299:
2279:
2259:
2212:
1945:
1923:
1901:
1872:
1836:
1780:
1750:
1749:{\displaystyle c=a+bi}
1715:
1714:{\displaystyle c^{*}c}
1621:
1583:
1400:
1327:
1294:
1166:s have raised indices?
1156:
1071:
983:
903:s have raised indices?
893:
809:
725:
643:
565:
542:
466:
443:
350:
327:
221:
3152:
3019:
2916:
2641:
2595:
2441:
2439:{\displaystyle v^{t}}
2414:
2412:{\displaystyle v^{T}}
2388:
2356:
2320:
2300:
2280:
2260:
2213:
1946:
1924:
1902:
1873:
1837:
1781:
1751:
1721:on a complex number,
1716:
1657:defecates scalars. --
1622:
1584:
1401:
1328:
1295:
1157:
1079:use any bilinear form
1072:
984:
927:Transpose and tensors
894:
817:use any bilinear form
810:
726:
644:
566:
543:
467:
444:
390:to be the linear map
351:
328:
274:to be the linear map
222:
36:level-5 vital article
3063:
2925:
2884:
2604:
2558:
2423:
2396:
2364:
2332:
2309:
2289:
2269:
2249:
2192:
1933:
1911:
1882:
1860:
1790:
1760:
1725:
1695:
1597:
1410:
1360:
1304:
1254:
1100:
1020:
947:
837:
758:
689:
665:hermitian transpose?
617:
553:
500:
454:
401:
338:
285:
187:
132:mathematics articles
3263:context is common.
3161:here on Knowledge.
2119:Orthogonal Matrices
1685:conjugate transpose
1548:
1199:matrix with either
3147:
3014:
2911:
2731:, yodalearning.com
2636:
2590:
2436:
2409:
2383:
2382:
2351:
2350:
2315:
2295:
2275:
2255:
2208:
2076:. So the quantity
1941:
1919:
1897:
1868:
1832:
1776:
1746:
1711:
1617:
1616:
1615:
1579:
1578:
1534:
1396:
1395:
1394:
1323:
1290:
1152:
1067:
979:
889:
805:
721:
639:
561:
538:
537:
462:
439:
438:
346:
323:
322:
217:
173:Wile E. Heresiarch
101:Mathematics portal
45:content assessment
3198:⟨Mv,w⟩ = ⟨v,Mw⟩
2784:
2768:comment added by
2652:comment added by
2634:
2496:hermitian adjoint
2278:{\displaystyle w}
2258:{\displaystyle o}
2131:as one for which
2112:
1689:complex conjugate
1354:orthonormal basis
1150:
977:
887:
719:
613:What about using
557:
458:
342:
215:
166:
165:
162:
161:
158:
157:
3330:
3190:Serious omission
3156:
3154:
3153:
3148:
3128:
3127:
3097:
3096:
3075:
3074:
3058:
3056:
3055:
3038:
3023:
3021:
3020:
3015:
3013:
2990:
2989:
2959:
2958:
2937:
2936:
2920:
2918:
2917:
2912:
2910:
2909:
2898:
2869:
2664:
2645:
2643:
2642:
2637:
2635:
2630:
2625:
2620:
2619:
2618:
2612:
2599:
2597:
2596:
2591:
2589:
2588:
2583:
2574:
2573:
2572:
2566:
2502:
2445:
2443:
2442:
2437:
2435:
2434:
2418:
2416:
2415:
2410:
2408:
2407:
2392:
2390:
2389:
2384:
2378:
2377:
2376:
2360:
2358:
2357:
2352:
2346:
2345:
2344:
2324:
2322:
2321:
2316:
2304:
2302:
2301:
2296:
2284:
2282:
2281:
2276:
2264:
2262:
2261:
2256:
2217:
2215:
2214:
2209:
2204:
2203:
2111:
2109:
2101:
2095:
2047:
2032:
1996:
1950:
1948:
1947:
1942:
1940:
1928:
1926:
1925:
1920:
1918:
1906:
1904:
1903:
1898:
1896:
1895:
1890:
1877:
1875:
1874:
1869:
1867:
1841:
1839:
1838:
1833:
1831:
1830:
1818:
1817:
1802:
1801:
1785:
1783:
1782:
1777:
1772:
1771:
1755:
1753:
1752:
1747:
1720:
1718:
1717:
1712:
1707:
1706:
1626:
1624:
1623:
1618:
1613:
1612:
1588:
1586:
1585:
1580:
1571:
1570:
1558:
1557:
1547:
1542:
1530:
1529:
1517:
1516:
1507:
1506:
1488:
1487:
1475:
1474:
1465:
1464:
1446:
1445:
1436:
1435:
1426:
1425:
1405:
1403:
1402:
1397:
1392:
1391:
1376:
1375:
1332:
1330:
1329:
1324:
1316:
1315:
1299:
1297:
1296:
1291:
1289:
1288:
1279:
1278:
1269:
1268:
1161:
1159:
1158:
1153:
1151:
1149:
1148:
1139:
1138:
1129:
1128:
1116:
1076:
1074:
1073:
1068:
1066:
1065:
1047:
1046:
988:
986:
985:
980:
978:
973:
972:
963:
898:
896:
895:
890:
888:
886:
885:
876:
875:
866:
865:
853:
814:
812:
811:
806:
804:
803:
785:
784:
730:
728:
727:
722:
720:
715:
714:
705:
648:
646:
645:
640:
629:
628:
570:
568:
567:
562:
556:
547:
545:
544:
539:
514:
513:
512:
471:
469:
468:
463:
457:
448:
446:
445:
440:
424:
423:
422:
355:
353:
352:
347:
341:
332:
330:
329:
324:
296:
295:
290:
226:
224:
223:
218:
216:
214:
210:
204:
197:
191:
169:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
3338:
3337:
3333:
3332:
3331:
3329:
3328:
3327:
3278:
3277:
3240:
3199:
3192:
3066:
3061:
3060:
3053:
3051:
3043:
3028:
3006:
2928:
2923:
2922:
2893:
2882:
2881:
2855:
2845:
2803:adjugate matrix
2790:
2760:
2722:
2683:
2647:
2607:
2602:
2601:
2578:
2561:
2556:
2555:
2552:
2500:
2467:In the section
2465:
2426:
2421:
2420:
2399:
2394:
2393:
2369:
2362:
2361:
2337:
2330:
2329:
2307:
2306:
2287:
2286:
2267:
2266:
2247:
2246:
2244:
2225:—Ben FrantzDale
2195:
2190:
2189:
2121:
2103:
2099:
2096:
2034:
2020:
2018:
2011:
1987:
1931:
1930:
1909:
1908:
1885:
1880:
1879:
1858:
1857:
1844:—Ben FrantzDale
1822:
1809:
1793:
1788:
1787:
1763:
1758:
1757:
1723:
1722:
1698:
1693:
1692:
1601:
1595:
1594:
1562:
1549:
1521:
1508:
1495:
1479:
1466:
1453:
1437:
1427:
1414:
1408:
1407:
1380:
1364:
1358:
1357:
1335:—Ben FrantzDale
1307:
1302:
1301:
1280:
1270:
1257:
1252:
1251:
1175:—Ben FrantzDale
1140:
1130:
1117:
1098:
1097:
1057:
1038:
1018:
1017:
964:
945:
944:
929:
924:
909:—Ben FrantzDale
877:
867:
854:
835:
834:
795:
776:
756:
755:
706:
687:
686:
679:
667:
651:—Ben FrantzDale
620:
615:
614:
551:
550:
548:
503:
498:
497:
452:
451:
449:
413:
399:
398:
336:
335:
333:
288:
283:
282:
240:
205:
192:
185:
184:
181:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
3336:
3334:
3326:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3280:
3279:
3276:
3275:
3239:
3236:
3197:
3191:
3188:
3146:
3143:
3140:
3137:
3134:
3131:
3126:
3123:
3120:
3115:
3112:
3109:
3106:
3103:
3100:
3095:
3092:
3089:
3084:
3081:
3078:
3073:
3069:
3059:it is used as
3012:
3009:
3005:
3002:
2999:
2996:
2993:
2988:
2985:
2982:
2977:
2974:
2971:
2968:
2965:
2962:
2957:
2954:
2951:
2946:
2943:
2940:
2935:
2931:
2908:
2905:
2902:
2897:
2892:
2889:
2844:
2841:
2799:
2798:
2789:
2786:
2759:
2756:
2733:
2732:
2721:
2718:
2686:(also written
2682:
2679:
2678:
2677:
2633:
2629:
2623:
2617:
2611:
2587:
2582:
2577:
2571:
2565:
2551:
2548:
2547:
2546:
2545:
2544:
2528:
2513:
2512:
2501:Sławomir Biały
2464:
2461:
2433:
2429:
2406:
2402:
2381:
2375:
2371:
2349:
2343:
2339:
2314:
2294:
2274:
2254:
2243:
2240:
2238:
2236:
2235:
2207:
2202:
2198:
2120:
2117:
2016:
2009:
1972:
1971:
1970:
1969:
1968:
1967:
1939:
1917:
1894:
1889:
1866:
1829:
1825:
1821:
1816:
1812:
1808:
1805:
1800:
1796:
1775:
1770:
1766:
1745:
1742:
1739:
1736:
1733:
1730:
1710:
1705:
1701:
1681:
1678:Talk:Transpose
1670:
1669:
1611:
1608:
1604:
1577:
1574:
1569:
1565:
1561:
1556:
1552:
1546:
1541:
1537:
1533:
1528:
1524:
1520:
1515:
1511:
1505:
1502:
1498:
1494:
1491:
1486:
1482:
1478:
1473:
1469:
1463:
1460:
1456:
1452:
1449:
1444:
1440:
1434:
1430:
1424:
1421:
1417:
1390:
1387:
1383:
1379:
1374:
1371:
1367:
1350:
1349:
1348:
1347:
1346:
1345:
1322:
1319:
1314:
1310:
1287:
1283:
1277:
1273:
1267:
1264:
1260:
1239:
1186:
1185:
1170:
1169:
1168:
1167:
1147:
1143:
1137:
1133:
1127:
1124:
1120:
1114:
1111:
1108:
1105:
1090:
1083:
1082:
1064:
1060:
1056:
1053:
1050:
1045:
1041:
1037:
1034:
1031:
1028:
1025:
1012:
1011:
991:
990:
976:
971:
967:
961:
958:
955:
952:
928:
925:
923:
920:
905:
904:
884:
880:
874:
870:
864:
861:
857:
851:
848:
845:
842:
827:
802:
798:
794:
791:
788:
783:
779:
775:
772:
769:
766:
763:
718:
713:
709:
703:
700:
697:
694:
678:
675:
666:
663:
662:
661:
638:
635:
632:
627:
623:
610:
609:
575:
574:
573:
572:
560:
535:
532:
529:
526:
523:
520:
517:
511:
506:
496:
476:
475:
474:
473:
461:
436:
433:
430:
427:
421:
416:
412:
409:
406:
397:
360:
359:
358:
357:
345:
320:
317:
314:
311:
308:
305:
302:
299:
294:
281:
239:
236:
228:
213:
209:
203:
200:
196:
180:
177:
168:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
3335:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3285:
3283:
3274:
3270:
3266:
3262:
3257:
3256:
3255:
3254:
3250:
3246:
3237:
3235:
3234:
3230:
3226:
3222:
3218:
3214:
3209:
3207:
3202:
3196:
3189:
3187:
3186:
3182:
3178:
3174:
3169:
3167:
3166:pertransposed
3164:I needed the
3162:
3160:
3141:
3138:
3135:
3132:
3129:
3113:
3110:
3107:
3104:
3101:
3098:
3079:
3076:
3071:
3067:
3049:
3048:
3040:
3036:
3032:
3025:
3010:
3003:
3000:
2997:
2994:
2991:
2975:
2972:
2969:
2966:
2963:
2960:
2941:
2938:
2933:
2929:
2906:
2903:
2900:
2890:
2887:
2878:
2876:
2875:pertransposed
2871:
2867:
2863:
2859:
2852:
2850:
2849:pertransposed
2843:Pertransposed
2842:
2840:
2839:
2835:
2831:
2827:
2822:
2817:
2816:
2812:
2808:
2804:
2795:
2794:
2793:
2787:
2785:
2783:
2779:
2775:
2771:
2767:
2757:
2755:
2754:
2750:
2746:
2742:
2738:
2730:
2727:
2726:
2725:
2719:
2717:
2716:
2712:
2708:
2703:
2701:
2697:
2693:
2689:
2680:
2676:
2673:
2672:
2667:
2666:
2665:
2663:
2659:
2655:
2651:
2621:
2585:
2575:
2549:
2543:
2540:
2539:
2534:
2529:
2527:
2524:
2523:
2517:
2516:
2515:
2514:
2511:
2507:
2503:
2497:
2492:
2491:
2490:
2489:
2486:
2485:
2480:
2479:
2474:
2470:
2462:
2460:
2459:
2455:
2451:
2450:129.32.11.206
2447:
2431:
2427:
2404:
2400:
2379:
2373:
2370:
2347:
2341:
2338:
2326:
2312:
2292:
2272:
2252:
2239:
2234:
2230:
2226:
2222:
2205:
2196:
2187:
2183:
2179:
2178:
2177:
2176:
2172:
2168:
2162:
2160:
2157:
2153:
2149:
2144:
2142:
2138:
2134:
2130:
2126:
2118:
2116:
2115:
2107:
2102:
2093:
2090:
2086:
2082:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2045:
2041:
2037:
2031:
2027:
2023:
2015:
2008:
2004:
2000:
1995:
1991:
1985:
1984:vector spaces
1981:
1977:
1966:
1962:
1958:
1954:
1953:Galois theory
1892:
1855:
1854:
1853:
1849:
1845:
1827:
1823:
1819:
1814:
1810:
1806:
1803:
1798:
1794:
1773:
1764:
1743:
1740:
1737:
1734:
1731:
1728:
1708:
1703:
1699:
1690:
1686:
1682:
1679:
1674:
1673:
1672:
1671:
1668:
1664:
1660:
1655:
1651:
1647:
1642:
1641:
1640:
1638:
1634:
1630:
1627:, is needed.
1609:
1606:
1602:
1592:
1575:
1572:
1563:
1559:
1554:
1550:
1539:
1535:
1531:
1526:
1522:
1513:
1509:
1503:
1500:
1496:
1489:
1484:
1480:
1471:
1467:
1461:
1458:
1454:
1447:
1442:
1438:
1432:
1428:
1422:
1419:
1415:
1388:
1385:
1381:
1377:
1372:
1369:
1365:
1355:
1344:
1340:
1336:
1320:
1317:
1308:
1285:
1281:
1275:
1271:
1265:
1262:
1258:
1249:
1244:
1240:
1236:
1235:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1189:
1188:
1187:
1184:
1180:
1176:
1172:
1171:
1165:
1145:
1141:
1135:
1131:
1125:
1122:
1118:
1112:
1106:
1095:
1094:metric tensor
1091:
1088:
1087:
1085:
1084:
1080:
1062:
1058:
1051:
1048:
1039:
1035:
1029:
1023:
1014:
1013:
1009:
1005:
1001:
997:
993:
992:
974:
965:
959:
953:
942:
938:
937:
936:
934:
926:
921:
919:
918:
914:
910:
902:
882:
878:
872:
868:
862:
859:
855:
849:
843:
832:
831:metric tensor
828:
825:
824:
823:
820:
818:
800:
796:
789:
786:
777:
773:
767:
761:
751:
749:
745:
741:
737:
732:
716:
707:
701:
695:
684:
676:
674:
672:
664:
660:
656:
652:
633:
621:
612:
611:
608:
604:
600:
595:
594:
593:
592:
589:
584:
580:
558:
533:
530:
527:
524:
518:
504:
495:
494:
493:
492:
491:
489:
485:
481:
459:
434:
431:
428:
425:
410:
404:
396:
395:
394:: W*→V* with
393:
389:
385:
381:
378:V and W with
377:
376:vector spaces
373:
369:
365:
364:
363:
343:
318:
315:
312:
309:
303:
297:
292:
280:
279:
278:: W*→V* with
277:
273:
269:
265:
262:V and W with
261:
260:vector spaces
257:
253:
249:
248:
247:
245:
237:
235:
234:
231:
198:
178:
176:
174:
153:
149:
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
3245:103.118.50.5
3241:
3220:
3216:
3212:
3211:The section
3210:
3205:
3203:
3200:
3193:
3173:pertranspose
3172:
3170:
3165:
3163:
3159:pertranspose
3158:
3052:. Retrieved
3046:
3041:
3034:
3030:
3026:
2879:
2874:
2872:
2857:
2853:
2848:
2846:
2825:
2818:
2800:
2791:
2764:— Preceding
2761:
2736:
2734:
2723:
2720:Yodalearning
2704:
2699:
2695:
2691:
2687:
2684:
2670:
2648:— Preceding
2553:
2537:
2521:
2483:
2477:
2466:
2448:
2327:
2245:
2237:
2185:
2181:
2163:
2158:
2155:
2151:
2147:
2145:
2140:
2136:
2132:
2128:
2124:
2122:
2091:
2084:
2080:
2077:
2073:
2069:
2065:
2061:
2053:
2049:
2043:
2039:
2035:
2029:
2025:
2021:
2013:
2006:
2002:
1993:
1989:
1979:
1975:
1973:
1677:
1653:
1649:
1645:
1351:
1223:matrices. --
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1163:
1007:
999:
940:
930:
906:
900:
821:
752:
747:
739:
733:
682:
680:
668:
582:
578:
576:
487:
483:
479:
477:
391:
387:
383:
367:
361:
275:
271:
267:
251:
243:
241:
182:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
34:
2072:is dual to
2058:dual spaces
1929:that keeps
588:Quuxplusone
380:dual spaces
370:: V→W is a
264:dual spaces
254:: V→W is a
123:Mathematics
114:mathematics
70:Mathematics
3282:Categories
3217:Especially
3054:2020-01-08
2707:JDAWiseman
2533:Dual basis
2150:such that
2056:* are the
1999:linear map
1004:dual space
931:I posed a
744:dual space
599:Cesiumfrog
549:for every
450:for every
372:linear map
334:for every
256:linear map
2860:: 77–82.
2737:Transpose
2473:this edit
1629:Bo Jacoby
1352:Using an
384:transpose
268:transpose
39:is rated
3050:(master)
3042:But, in
2851:matrix.
2788:Adjugate
2778:contribs
2770:Lehnekbn
2766:unsigned
2741:linkspam
2650:unsigned
2167:Horn.imh
2089:covector
1173:Thanks.
907:Thanks.
374:between
258:between
230:yanneman
3265:Rgdboer
3221:defined
3213:Adjoint
3206:Adjoint
2830:Rgdboer
2821:WP:User
2807:Rgdboer
2745:Rgdboer
2671:Quondum
2554:States
2538:Quondum
2522:Quondum
2484:Quondum
2129:defined
2087:by the
2012:} and {
1982:be two
1406:, and
242:In the
150:on the
41:C-class
2826:Devssh
2100:Dr Dec
2052:* and
2048:where
2001:. Let
1986:, and
1591:tensor
1248:kludge
1203:=1 or
581:" or "
571:in W*.
472:in W*.
356:in W*.
170:Note =
47:scale.
2685:: -->
2535:. —
1997:be a
1957:Tango
1243:mixex
1225:Tango
671:Moala
28:This
3269:talk
3249:talk
3229:talk
3181:talk
3037:(1).
2834:talk
2819:New
2811:talk
2774:talk
2749:talk
2743:. —
2711:talk
2658:talk
2506:talk
2454:talk
2325:...
2229:talk
2171:talk
2154:and
2106:Talk
2064:and
2042:* →
2033:and
1988:ƒ :
1978:and
1974:Let
1961:talk
1848:talk
1663:talk
1633:talk
1339:talk
1229:talk
1179:talk
1086:So:
913:talk
822:So:
655:talk
603:talk
2862:doi
2698:or
2690:′,
2498:.)
2419:or
2221:QED
2127:is
2097:~~
2060:of
1659:pma
1219:by
1211:by
1195:by
1006:of
943:as
746:of
685:as
386:of
366:If
270:of
250:If
142:Mid
3284::
3271:)
3251:)
3231:)
3223:.
3183:)
3177:TN
3133:−
3102:−
3039:.
3033:.
2995:−
2964:−
2904:×
2891:∈
2870:.
2836:)
2813:)
2780:)
2776:•
2751:)
2713:)
2702:)
2694:,
2660:)
2632:¯
2586:∗
2508:)
2456:)
2313:ω
2293:ω
2231:)
2223:.
2201:⊤
2173:)
2165:--
2152:GG
2110:~~
2094:.
2038::
2028:→
2024::
1992:→
1963:)
1850:)
1799:∗
1769:⊤
1704:∗
1665:)
1639:.
1635:)
1568:⊤
1545:⊤
1497:δ
1382:δ
1356:,
1341:)
1313:⊤
1231:)
1181:)
1110:‖
1104:‖
1096::
1063:∗
1055:→
1044:⊤
970:⊤
957:‖
951:‖
915:)
847:‖
841:‖
833::
801:∗
793:→
782:⊤
731:.
712:⊤
699:‖
693:‖
657:)
649:?
634:ϕ
626:⊤
605:)
559:ϕ
531:∘
528:ϕ
519:ϕ
460:ϕ
432:∘
429:ϕ
411:ϕ
344:ϕ
316:∘
313:ϕ
304:ϕ
212:Θ
202:Θ
3267:(
3247:(
3227:(
3179:(
3145:)
3142:1
3139::
3136:1
3130::
3125:d
3122:n
3119:e
3114:,
3111:1
3108::
3105:1
3099::
3094:d
3091:n
3088:e
3083:(
3080:A
3077:=
3072:P
3068:A
3057:.
3035:9
3011:′
3008:)
3004:1
3001::
2998:1
2992::
2987:d
2984:n
2981:e
2976:,
2973:1
2970::
2967:1
2961::
2956:d
2953:n
2950:e
2945:(
2942:A
2939:=
2934:P
2930:A
2907:n
2901:n
2896:R
2888:A
2868:.
2864::
2832:(
2809:(
2772:(
2747:(
2709:(
2700:A
2696:A
2692:A
2688:A
2656:(
2628:A
2622:=
2616:T
2610:A
2581:A
2576:=
2570:T
2564:A
2504:(
2452:(
2432:t
2428:v
2405:T
2401:v
2380:v
2374:T
2348:v
2342:t
2273:w
2253:o
2227:(
2206:G
2197:G
2186:G
2182:G
2169:(
2159:G
2156:G
2148:G
2141:G
2137:G
2135:=
2133:G
2125:G
2108:)
2104:(
2092:v
2085:v
2081:v
2078:v
2074:v
2070:v
2066:W
2062:V
2054:W
2050:V
2046:*
2044:V
2040:W
2036:F
2030:W
2026:V
2022:F
2017:j
2014:w
2010:i
2007:v
2003:F
1994:W
1990:V
1980:W
1976:V
1959:(
1938:R
1916:C
1893:n
1888:C
1865:C
1846:(
1828:2
1824:b
1820:+
1815:2
1811:a
1807:=
1804:c
1795:c
1774:v
1765:v
1744:i
1741:b
1738:+
1735:a
1732:=
1729:c
1709:c
1700:c
1661:(
1654:R
1650:R
1646:R
1631:(
1610:j
1607:i
1603:g
1576:.
1573:v
1564:v
1560:=
1555:j
1551:v
1540:j
1536:v
1532:=
1527:j
1523:v
1519:)
1514:i
1510:v
1504:j
1501:i
1493:(
1490:=
1485:j
1481:v
1477:)
1472:i
1468:v
1462:j
1459:i
1455:g
1451:(
1448:=
1443:j
1439:v
1433:i
1429:v
1423:j
1420:i
1416:g
1389:j
1386:i
1378:=
1373:j
1370:i
1366:g
1337:(
1321:v
1318:g
1309:v
1286:j
1282:v
1276:i
1272:v
1266:j
1263:i
1259:g
1227:(
1221:n
1217:m
1213:m
1209:n
1205:m
1201:n
1197:m
1193:n
1177:(
1164:v
1146:j
1142:v
1136:i
1132:v
1126:j
1123:i
1119:g
1113:=
1107:v
1059:V
1052:V
1049::
1040:v
1036:=
1033:)
1030:v
1027:(
1024:f
1008:c
1000:c
989:.
975:v
966:v
960:=
954:v
941:v
911:(
901:v
883:j
879:v
873:i
869:v
863:j
860:i
856:g
850:=
844:v
797:V
790:V
787::
778:v
774:=
771:)
768:v
765:(
762:f
748:c
740:c
717:v
708:v
702:=
696:v
683:v
653:(
637:)
631:(
622:f
601:(
583:f
579:f
534:f
525:=
522:)
516:(
510:T
505:f
488:φ
486:(
484:f
480:f
435:f
426:=
420:T
415:)
408:(
405:f
392:f
388:f
368:f
319:f
310:=
307:)
301:(
298:f
293:t
276:f
272:f
252:f
208:d
199:A
195:d
154:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
