2740:
2490:
1511:
2735:{\displaystyle \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N} }
1697:
on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the
174:. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be
1252:
2108:
2373:
671:
1006:
820:
161:
879:
1095:
2479:
481:
412:
726:
1165:
343:
53:. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an
1982:
2114:
1625:
1580:
1193:
1129:
914:
1506:{\displaystyle \forall {\textbf {a}}:=;\ \|a_{1}{\textbf {e}}_{1}+a_{2}{\textbf {e}}_{2}+\cdots +a_{n}{\textbf {e}}_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}}
1939:
1910:
1804:
581:
540:
514:
1974:
1203:
Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of
586:
926:
742:
1693:, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits
118:
2845:
2815:
826:
1023:
2401:
729:
77:
of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the
884:
It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.
418:
349:
2868:
676:
2782:
78:
1138:
281:
2103:{\displaystyle (1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}}
2863:
2395:
to be the space of all real-valued functions continuous on the interval and taking the inner product to be
57:
if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a
1942:
1876:
170:
deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of
58:
2752:
is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that
1219:
Orthonormal sets have certain very appealing properties, which make them particularly easy to work with.
1204:
2368:{\displaystyle (2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left^{\frac {1}{2}}=\left^{\frac {1}{2}}=1.}
1686:
1533:
1519:
1606:
1561:
1174:
1103:
895:
2829:
917:
39:
2837:
1682:
550:= 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the
105:
101:
2841:
2811:
2807:
2777:
1915:
1886:
487:
82:
62:
1789:
1720:
1699:
2803:
1690:
1132:
97:
and specifying that two vectors in the plane are orthogonal if their dot product is zero.
560:
519:
493:
2388:
2384:
1716:
1015:
167:
31:
1947:
2857:
2795:
1168:
74:
47:
2757:
666:{\displaystyle \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0}
35:
17:
1001:{\displaystyle \left\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}}
815:{\displaystyle \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)}
1872:
551:
171:
94:
90:
50:
1694:
1208:
2833:
733:
1842:
where i≠j are orthogonal, and all vectors are clearly of unit length. So {
112:
of a vector is the square root of the vector dotted with itself. That is,
104:
of a vector is motivated by a desire to extend the intuitive notion of the
186:
What does a pair of orthonormal vectors in 2-D Euclidean space look like?
1880:
156:{\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}}
93:). This definition can be formalized in Cartesian space by defining the
1558:} is a linearly independent list of vectors in an inner-product space
874:{\displaystyle \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}}
27:
Property of two or more vectors that are orthogonal and of unit length
2387:
is a method of expressing a periodic function in terms of sinusoidal
1090:{\displaystyle \forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}}
108:
of a vector to higher-dimensional spaces. In
Cartesian space, the
2474:{\displaystyle \langle f,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx}
1883:
inner product is assumed unless otherwise stated. Two functions
1612:
1567:
1180:
993:
901:
1689:
elsewhere. The Gram-Schmidt theorem, together with the
860:
796:
89:
if the angle between them is 90° (i.e. if they form a
2493:
2404:
2117:
1985:
1950:
1918:
1889:
1792:
1609:
1564:
1255:
1177:
1141:
1106:
1026:
929:
898:
829:
745:
679:
589:
563:
522:
496:
421:
352:
284:
121:
476:{\displaystyle {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1}
407:{\displaystyle {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1}
2734:
2473:
2367:
2102:
1968:
1933:
1904:
1798:
1619:
1574:
1505:
1187:
1159:
1123:
1089:
1000:
908:
873:
814:
721:{\displaystyle \tan \theta _{1}=-\cot \theta _{2}}
720:
665:
575:
534:
508:
475:
406:
337:
155:
2748:However, this is of little consequence, because
1160:{\displaystyle \langle \cdot ,\cdot \rangle }
338:{\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\quad }
8:
2417:
2405:
2026:
1996:
1398:
1310:
1154:
1142:
1068:
1042:
130:
122:
1249:} is an orthonormal list of vectors, then
2728:
2727:
2683:
2645:
2616:
2584:
2546:
2517:
2499:
2492:
2435:
2427:
2403:
2348:
2331:
2326:
2308:
2302:
2297:
2273:
2256:
2251:
2233:
2227:
2222:
2203:
2198:
2192:
2175:
2170:
2161:
2156:
2150:
2133:
2128:
2116:
2088:
2087:
2041:
2036:
1984:
1949:
1917:
1888:
1791:
1611:
1610:
1608:
1582:, then there exists an orthonormal list {
1566:
1565:
1563:
1497:
1492:
1485:
1476:
1461:
1456:
1449:
1440:
1431:
1426:
1419:
1410:
1401:
1391:
1385:
1384:
1377:
1358:
1352:
1351:
1344:
1331:
1325:
1324:
1317:
1295:
1276:
1260:
1259:
1254:
1179:
1178:
1176:
1140:
1120:
1111:
1105:
1078:
1062:
1049:
1025:
992:
991:
971:
952:
939:
928:
900:
899:
897:
859:
850:
837:
828:
795:
786:
759:
744:
712:
690:
678:
651:
635:
616:
600:
588:
562:
521:
495:
459:
452:
447:
437:
430:
425:
422:
420:
390:
383:
378:
368:
361:
356:
353:
351:
322:
312:
299:
289:
283:
275:Expanding these terms gives 3 equations:
146:
138:
136:
125:
120:
1518:. Every orthonormal list of vectors is
2764:and therefore an orthonormal basis of
1681:Proof of the Gram-Schmidt theorem is
7:
2826:Fundamentals of Circuits and Filters
263:From the unit length restriction on
252:From the unit length restriction on
241:From the orthogonality restriction,
1386:
1353:
1326:
1261:
100:Similarly, the construction of the
2802:(2nd ed.), Berlin, New York:
2095:
2092:
2089:
1256:
1027:
25:
213:). Consider the restrictions on x
147:
139:
126:
2720:
2127:
2086:
1995:
334:
2701:
2692:
2663:
2654:
2631:
2625:
2602:
2593:
2564:
2555:
2532:
2526:
2462:
2456:
2450:
2444:
2327:
2322:
2316:
2309:
2252:
2247:
2241:
2234:
2199:
2193:
2189:
2183:
2176:
2171:
2157:
2151:
2147:
2141:
2134:
2129:
2124:
2118:
2068:
2062:
2056:
2050:
2023:
2017:
2008:
2002:
1992:
1986:
1963:
1951:
1928:
1922:
1899:
1893:
1863:} forms an orthonormal basis.
1620:{\displaystyle {\mathcal {V}}}
1575:{\displaystyle {\mathcal {V}}}
1493:
1477:
1457:
1441:
1427:
1411:
1301:
1269:
1188:{\displaystyle {\mathcal {V}}}
1124:{\displaystyle \delta _{ij}\,}
909:{\displaystyle {\mathcal {V}}}
830:
765:
752:
570:
564:
542:immediately gives the result r
529:
523:
503:
497:
1:
557:After substitution, Equation
486:Converting from Cartesian to
2783:Orthonormal function system
490:, and considering Equation
2885:
2745:forms an orthonormal set.
237:form an orthonormal pair.
166:Many important results in
2800:Linear Algebra Done Right
1941:are orthonormal over the
1755:} where
2756:be finite makes the set
1934:{\displaystyle \psi (x)}
1905:{\displaystyle \phi (x)}
1799:{\displaystyle \vdots }
2824:Chen, Wai-Kai (2009),
2736:
2475:
2369:
2104:
1970:
1935:
1906:
1800:
1621:
1576:
1507:
1189:
1161:
1125:
1091:
1002:
910:
875:
816:
730:trigonometric identity
722:
667:
577:
536:
510:
477:
408:
339:
157:
2737:
2484:it can be shown that
2476:
2370:
2105:
1971:
1936:
1907:
1867:Real-valued functions
1801:
1622:
1577:
1508:
1190:
1162:
1126:
1092:
1003:
911:
876:
817:
723:
668:
578:
537:
511:
478:
409:
340:
158:
2491:
2402:
2115:
1983:
1948:
1916:
1887:
1790:
1607:
1562:
1534:Gram-Schmidt theorem
1520:linearly independent
1253:
1175:
1139:
1104:
1024:
927:
920:. A set of vectors
896:
827:
743:
677:
673:. Rearranging gives
587:
561:
520:
494:
419:
350:
282:
119:
73:The construction of
2869:Functional analysis
2440:
2307:
2232:
2046:
1687:discussed at length
918:inner-product space
576:{\displaystyle (1)}
535:{\displaystyle (3)}
509:{\displaystyle (2)}
40:inner product space
18:Orthonormal vectors
2732:
2471:
2423:
2391:functions. Taking
2365:
2293:
2218:
2100:
2032:
1966:
1931:
1902:
1871:When referring to
1813:
1796:
1772:
1758:
1617:
1572:
1503:
1211:on vector spaces.
1185:
1157:
1121:
1087:
998:
906:
871:
869:
812:
805:
718:
663:
573:
532:
506:
473:
404:
335:
153:
69:Intuitive overview
2847:978-1-4200-5887-1
2817:978-0-387-98258-8
2778:Orthogonalization
2710:
2709:
2672:
2671:
2640:
2639:
2611:
2610:
2573:
2572:
2541:
2540:
2512:
2511:
2356:
2281:
1824:
1823:
1820:= (0, 0, ..., 1)
1779:= (0, 1, ..., 0)
1765:= (1, 0, ..., 0)
1388:
1355:
1328:
1309:
1263:
1205:diagonalizability
868:
804:
488:polar coordinates
465:
396:
229:required to make
151:
63:orthonormal basis
16:(Redirected from
2876:
2850:
2828:(3rd ed.),
2820:
2741:
2739:
2738:
2733:
2731:
2716:
2712:
2711:
2705:
2704:
2684:
2673:
2667:
2666:
2646:
2641:
2635:
2634:
2617:
2612:
2606:
2605:
2585:
2574:
2568:
2567:
2547:
2542:
2536:
2535:
2518:
2513:
2504:
2500:
2480:
2478:
2477:
2472:
2439:
2434:
2374:
2372:
2371:
2366:
2358:
2357:
2349:
2347:
2343:
2336:
2335:
2330:
2312:
2306:
2301:
2283:
2282:
2274:
2272:
2268:
2261:
2260:
2255:
2237:
2231:
2226:
2208:
2207:
2202:
2196:
2179:
2174:
2166:
2165:
2160:
2154:
2137:
2132:
2109:
2107:
2106:
2101:
2099:
2098:
2045:
2040:
1975:
1973:
1972:
1969:{\displaystyle }
1967:
1940:
1938:
1937:
1932:
1911:
1909:
1908:
1903:
1828:Any two vectors
1805:
1803:
1802:
1797:
1731:
1730:
1721:coordinate space
1700:Spectral Theorem
1626:
1624:
1623:
1618:
1616:
1615:
1603:} of vectors in
1581:
1579:
1578:
1573:
1571:
1570:
1512:
1510:
1509:
1504:
1502:
1501:
1496:
1490:
1489:
1480:
1466:
1465:
1460:
1454:
1453:
1444:
1436:
1435:
1430:
1424:
1423:
1414:
1406:
1405:
1396:
1395:
1390:
1389:
1382:
1381:
1363:
1362:
1357:
1356:
1349:
1348:
1336:
1335:
1330:
1329:
1322:
1321:
1307:
1300:
1299:
1281:
1280:
1265:
1264:
1194:
1192:
1191:
1186:
1184:
1183:
1166:
1164:
1163:
1158:
1130:
1128:
1127:
1122:
1119:
1118:
1096:
1094:
1093:
1088:
1086:
1085:
1067:
1066:
1054:
1053:
1007:
1005:
1004:
999:
997:
996:
987:
983:
976:
975:
957:
956:
944:
943:
915:
913:
912:
907:
905:
904:
880:
878:
877:
872:
870:
861:
855:
854:
842:
841:
821:
819:
818:
813:
811:
807:
806:
797:
791:
790:
764:
763:
727:
725:
724:
719:
717:
716:
695:
694:
672:
670:
669:
664:
656:
655:
640:
639:
621:
620:
605:
604:
582:
580:
579:
574:
541:
539:
538:
533:
515:
513:
512:
507:
482:
480:
479:
474:
466:
464:
463:
458:
457:
456:
442:
441:
436:
435:
434:
423:
413:
411:
410:
405:
397:
395:
394:
389:
388:
387:
373:
372:
367:
366:
365:
354:
344:
342:
341:
336:
327:
326:
317:
316:
304:
303:
294:
293:
162:
160:
159:
154:
152:
150:
142:
137:
129:
21:
2884:
2883:
2879:
2878:
2877:
2875:
2874:
2873:
2854:
2853:
2848:
2823:
2818:
2804:Springer-Verlag
2794:
2791:
2774:
2685:
2647:
2618:
2586:
2548:
2519:
2498:
2494:
2489:
2488:
2400:
2399:
2381:
2325:
2292:
2288:
2287:
2250:
2217:
2213:
2212:
2197:
2155:
2113:
2112:
1981:
1980:
1946:
1945:
1914:
1913:
1885:
1884:
1869:
1862:
1855:
1848:
1841:
1834:
1819:
1806:
1788:
1787:
1778:
1764:
1754:
1747:
1740:
1713:
1708:
1691:axiom of choice
1676:
1669:
1662:
1651:
1644:
1637:
1605:
1604:
1602:
1595:
1588:
1560:
1559:
1557:
1550:
1543:
1529:
1491:
1481:
1455:
1445:
1425:
1415:
1397:
1383:
1373:
1350:
1340:
1323:
1313:
1291:
1272:
1251:
1250:
1248:
1239:
1232:
1217:
1201:
1173:
1172:
1137:
1136:
1133:Kronecker delta
1107:
1102:
1101:
1074:
1058:
1045:
1022:
1021:
967:
948:
935:
934:
930:
925:
924:
894:
893:
890:
846:
833:
825:
824:
782:
781:
777:
755:
741:
740:
732:to convert the
708:
686:
675:
674:
647:
631:
612:
596:
585:
584:
559:
558:
549:
545:
518:
517:
492:
491:
448:
446:
426:
424:
417:
416:
379:
377:
357:
355:
348:
347:
318:
308:
295:
285:
280:
279:
228:
224:
220:
216:
212:
208:
200:
196:
184:
117:
116:
85:are said to be
79:Cartesian plane
71:
55:orthonormal set
28:
23:
22:
15:
12:
11:
5:
2882:
2880:
2872:
2871:
2866:
2864:Linear algebra
2856:
2855:
2852:
2851:
2846:
2821:
2816:
2796:Axler, Sheldon
2790:
2787:
2786:
2785:
2780:
2773:
2770:
2743:
2742:
2730:
2726:
2723:
2719:
2715:
2708:
2703:
2700:
2697:
2694:
2691:
2688:
2682:
2679:
2676:
2670:
2665:
2662:
2659:
2656:
2653:
2650:
2644:
2638:
2633:
2630:
2627:
2624:
2621:
2615:
2609:
2604:
2601:
2598:
2595:
2592:
2589:
2583:
2580:
2577:
2571:
2566:
2563:
2560:
2557:
2554:
2551:
2545:
2539:
2534:
2531:
2528:
2525:
2522:
2516:
2510:
2507:
2503:
2497:
2482:
2481:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2438:
2433:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2407:
2385:Fourier series
2380:
2379:Fourier series
2377:
2376:
2375:
2364:
2361:
2355:
2352:
2346:
2342:
2339:
2334:
2329:
2324:
2321:
2318:
2315:
2311:
2305:
2300:
2296:
2291:
2286:
2280:
2277:
2271:
2267:
2264:
2259:
2254:
2249:
2246:
2243:
2240:
2236:
2230:
2225:
2221:
2216:
2211:
2206:
2201:
2195:
2191:
2188:
2185:
2182:
2178:
2173:
2169:
2164:
2159:
2153:
2149:
2146:
2143:
2140:
2136:
2131:
2126:
2123:
2120:
2110:
2097:
2094:
2091:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2044:
2039:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1994:
1991:
1988:
1965:
1962:
1959:
1956:
1953:
1930:
1927:
1924:
1921:
1901:
1898:
1895:
1892:
1879:, usually the
1868:
1865:
1860:
1853:
1846:
1839:
1832:
1826:
1825:
1822:
1821:
1817:
1811:
1808:
1807:
1795:
1786:
1784:
1781:
1780:
1776:
1770:
1767:
1766:
1762:
1756:
1752:
1745:
1738:
1717:standard basis
1712:
1711:Standard basis
1709:
1707:
1704:
1679:
1678:
1674:
1667:
1660:
1649:
1642:
1635:
1614:
1600:
1593:
1586:
1569:
1555:
1548:
1541:
1528:
1525:
1524:
1523:
1513:
1500:
1495:
1488:
1484:
1479:
1475:
1472:
1469:
1464:
1459:
1452:
1448:
1443:
1439:
1434:
1429:
1422:
1418:
1413:
1409:
1404:
1400:
1394:
1380:
1376:
1372:
1369:
1366:
1361:
1347:
1343:
1339:
1334:
1320:
1316:
1312:
1306:
1303:
1298:
1294:
1290:
1287:
1284:
1279:
1275:
1271:
1268:
1258:
1244:
1237:
1230:
1216:
1213:
1200:
1197:
1182:
1156:
1153:
1150:
1147:
1144:
1117:
1114:
1110:
1098:
1097:
1084:
1081:
1077:
1073:
1070:
1065:
1061:
1057:
1052:
1048:
1044:
1041:
1038:
1035:
1032:
1029:
1016:if and only if
1009:
1008:
995:
990:
986:
982:
979:
974:
970:
966:
963:
960:
955:
951:
947:
942:
938:
933:
903:
889:
886:
882:
881:
867:
864:
858:
853:
849:
845:
840:
836:
832:
822:
810:
803:
800:
794:
789:
785:
780:
776:
773:
770:
767:
762:
758:
754:
751:
748:
715:
711:
707:
704:
701:
698:
693:
689:
685:
682:
662:
659:
654:
650:
646:
643:
638:
634:
630:
627:
624:
619:
615:
611:
608:
603:
599:
595:
592:
572:
569:
566:
547:
543:
531:
528:
525:
505:
502:
499:
484:
483:
472:
469:
462:
455:
451:
445:
440:
433:
429:
414:
403:
400:
393:
386:
382:
376:
371:
364:
360:
345:
333:
330:
325:
321:
315:
311:
307:
302:
298:
292:
288:
273:
272:
261:
250:
226:
222:
218:
214:
210:
206:
198:
194:
183:
182:Simple example
180:
168:linear algebra
164:
163:
149:
145:
141:
135:
132:
128:
124:
70:
67:
32:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2881:
2870:
2867:
2865:
2862:
2861:
2859:
2849:
2843:
2839:
2835:
2831:
2827:
2822:
2819:
2813:
2809:
2805:
2801:
2797:
2793:
2792:
2788:
2784:
2781:
2779:
2776:
2775:
2771:
2769:
2767:
2763:
2759:
2755:
2751:
2746:
2724:
2721:
2717:
2713:
2706:
2698:
2695:
2689:
2686:
2680:
2677:
2674:
2668:
2660:
2657:
2651:
2648:
2642:
2636:
2628:
2622:
2619:
2613:
2607:
2599:
2596:
2590:
2587:
2581:
2578:
2575:
2569:
2561:
2558:
2552:
2549:
2543:
2537:
2529:
2523:
2520:
2514:
2508:
2505:
2501:
2495:
2487:
2486:
2485:
2468:
2465:
2459:
2453:
2447:
2441:
2436:
2431:
2428:
2424:
2420:
2414:
2411:
2408:
2398:
2397:
2396:
2394:
2390:
2386:
2378:
2362:
2359:
2353:
2350:
2344:
2340:
2337:
2332:
2319:
2313:
2303:
2298:
2294:
2289:
2284:
2278:
2275:
2269:
2265:
2262:
2257:
2244:
2238:
2228:
2223:
2219:
2214:
2209:
2204:
2186:
2180:
2167:
2162:
2144:
2138:
2121:
2111:
2083:
2080:
2077:
2074:
2071:
2065:
2059:
2053:
2047:
2042:
2037:
2033:
2029:
2020:
2014:
2011:
2005:
1999:
1989:
1979:
1978:
1977:
1960:
1957:
1954:
1944:
1925:
1919:
1896:
1890:
1882:
1878:
1874:
1866:
1864:
1859:
1852:
1845:
1838:
1831:
1816:
1812:
1810:
1809:
1793:
1785:
1783:
1782:
1775:
1771:
1769:
1768:
1761:
1757:
1751:
1744:
1737:
1733:
1732:
1729:
1728:
1727:
1725:
1722:
1718:
1710:
1705:
1703:
1701:
1696:
1692:
1688:
1684:
1673:
1666:
1659:
1655:
1648:
1641:
1634:
1630:
1599:
1592:
1585:
1554:
1547:
1540:
1536:
1535:
1531:
1530:
1526:
1521:
1517:
1514:
1498:
1486:
1482:
1473:
1470:
1467:
1462:
1450:
1446:
1437:
1432:
1420:
1416:
1407:
1402:
1392:
1378:
1374:
1370:
1367:
1364:
1359:
1345:
1341:
1337:
1332:
1318:
1314:
1304:
1296:
1292:
1288:
1285:
1282:
1277:
1273:
1266:
1247:
1243:
1236:
1229:
1225:
1222:
1221:
1220:
1214:
1212:
1210:
1206:
1198:
1196:
1171:defined over
1170:
1169:inner product
1151:
1148:
1145:
1134:
1115:
1112:
1108:
1082:
1079:
1075:
1071:
1063:
1059:
1055:
1050:
1046:
1039:
1036:
1033:
1030:
1020:
1019:
1018:
1017:
1014:
988:
984:
980:
977:
972:
968:
964:
961:
958:
953:
949:
945:
940:
936:
931:
923:
922:
921:
919:
887:
885:
865:
862:
856:
851:
847:
843:
838:
834:
823:
808:
801:
798:
792:
787:
783:
778:
774:
771:
768:
760:
756:
749:
746:
739:
738:
737:
735:
731:
713:
709:
705:
702:
699:
696:
691:
687:
683:
680:
660:
657:
652:
648:
644:
641:
636:
632:
628:
625:
622:
617:
613:
609:
606:
601:
597:
593:
590:
567:
555:
553:
526:
516:and Equation
500:
489:
470:
467:
460:
453:
449:
443:
438:
431:
427:
415:
401:
398:
391:
384:
380:
374:
369:
362:
358:
346:
331:
328:
323:
319:
313:
309:
305:
300:
296:
290:
286:
278:
277:
276:
270:
266:
262:
259:
255:
251:
248:
244:
240:
239:
238:
236:
232:
204:
192:
187:
181:
179:
177:
173:
169:
143:
133:
115:
114:
113:
111:
107:
103:
98:
96:
92:
88:
87:perpendicular
84:
80:
76:
75:orthogonality
68:
66:
64:
61:is called an
60:
56:
52:
49:
45:
41:
37:
33:
19:
2825:
2799:
2765:
2761:
2753:
2749:
2747:
2744:
2483:
2392:
2382:
1870:
1857:
1850:
1843:
1836:
1829:
1827:
1814:
1773:
1759:
1749:
1742:
1735:
1723:
1714:
1683:constructive
1680:
1671:
1664:
1657:
1653:
1646:
1639:
1632:
1628:
1597:
1590:
1583:
1552:
1545:
1538:
1532:
1515:
1245:
1241:
1234:
1227:
1223:
1218:
1202:
1199:Significance
1099:
1012:
1010:
891:
883:
556:
485:
274:
268:
264:
257:
253:
246:
242:
234:
230:
202:
190:
188:
185:
175:
165:
109:
99:
86:
72:
54:
51:unit vectors
46:if they are
43:
29:
1207:of certain
1013:orthonormal
736:term gives
552:unit circle
176:orthonormal
172:unit length
95:dot product
91:right angle
44:orthonormal
2858:Categories
2836:, p.
2830:Boca Raton
2806:, p.
1627:such that
1215:Properties
1011:is called
888:Definition
728:. Using a
48:orthogonal
2834:CRC Press
2725:∈
2707:π
2690:
2678:…
2669:π
2652:
2637:π
2623:
2608:π
2591:
2579:…
2570:π
2553:
2538:π
2524:
2509:π
2437:π
2432:π
2429:−
2425:∫
2418:⟩
2406:⟨
2314:ψ
2295:∫
2239:ϕ
2220:∫
2181:ψ
2139:ϕ
2060:ψ
2048:ϕ
2034:∫
2027:⟩
2015:ψ
2000:ϕ
1997:⟨
1920:ψ
1891:ϕ
1877:functions
1794:⋮
1695:operators
1527:Existence
1471:⋯
1399:‖
1368:⋯
1311:‖
1286:⋯
1257:∀
1209:operators
1155:⟩
1152:⋅
1146:⋅
1143:⟨
1109:δ
1076:δ
1069:⟩
1043:⟨
1028:∀
989:∈
981:…
962:…
863:π
848:θ
835:θ
831:⇒
799:π
784:θ
775:
757:θ
750:
734:cotangent
710:θ
706:
700:−
688:θ
684:
649:θ
645:
633:θ
629:
614:θ
610:
598:θ
594:
583:becomes
144:⋅
131:‖
123:‖
2798:(1997),
2772:See also
1943:interval
1875:-valued
1719:for the
1706:Examples
2808:106–110
2789:Sources
1537:. If {
1516:Theorem
1240:, ...,
1226:. If {
1224:Theorem
1167:is the
1131:is the
271:|| = 1.
260:|| = 1.
83:vectors
36:vectors
2844:
2814:
1685:, and
1308:
1100:where
916:be an
201:) and
106:length
81:, two
38:in an
34:, two
2758:dense
2389:basis
1856:,...,
1748:,...,
1670:,...,
1645:,...,
1596:,...,
1551:,...,
59:basis
2842:ISBN
2812:ISBN
2383:The
1912:and
1873:real
1715:The
1654:span
1652:) =
1629:span
1135:and
892:Let
267:, ||
256:, ||
249:= 0.
233:and
205:= (x
193:= (x
189:Let
110:norm
102:norm
42:are
2760:in
2687:cos
2649:cos
2620:cos
2588:sin
2550:sin
2521:sin
1976:if
1726:is
772:tan
747:tan
703:cot
681:tan
642:sin
626:sin
607:cos
591:cos
546:= r
225:, y
221:, y
217:, x
209:, y
197:, y
30:In
2860::
2840:,
2838:62
2832::
2810:,
2768:.
2363:1.
1881:L²
1849:,
1835:,
1741:,
1702:.
1677:).
1663:,
1638:,
1589:,
1544:,
1267::=
1233:,
1195:.
554:.
245:•
178:.
65:.
2766:C
2762:C
2754:n
2750:C
2729:N
2722:n
2718:,
2714:}
2702:)
2699:x
2696:n
2693:(
2681:,
2675:,
2664:)
2661:x
2658:2
2655:(
2643:,
2632:)
2629:x
2626:(
2614:,
2603:)
2600:x
2597:n
2594:(
2582:,
2576:,
2565:)
2562:x
2559:2
2556:(
2544:,
2533:)
2530:x
2527:(
2515:,
2506:2
2502:1
2496:{
2469:x
2466:d
2463:)
2460:x
2457:(
2454:g
2451:)
2448:x
2445:(
2442:f
2421:=
2415:g
2412:,
2409:f
2393:C
2360:=
2354:2
2351:1
2345:]
2341:x
2338:d
2333:2
2328:|
2323:)
2320:x
2317:(
2310:|
2304:b
2299:a
2290:[
2285:=
2279:2
2276:1
2270:]
2266:x
2263:d
2258:2
2253:|
2248:)
2245:x
2242:(
2235:|
2229:b
2224:a
2215:[
2210:=
2205:2
2200:|
2194:|
2190:)
2187:x
2184:(
2177:|
2172:|
2168:=
2163:2
2158:|
2152:|
2148:)
2145:x
2142:(
2135:|
2130:|
2125:)
2122:2
2119:(
2096:d
2093:n
2090:a
2084:,
2081:0
2078:=
2075:x
2072:d
2069:)
2066:x
2063:(
2057:)
2054:x
2051:(
2043:b
2038:a
2030:=
2024:)
2021:x
2018:(
2012:,
2009:)
2006:x
2003:(
1993:)
1990:1
1987:(
1964:]
1961:b
1958:,
1955:a
1952:[
1929:)
1926:x
1923:(
1900:)
1897:x
1894:(
1861:n
1858:e
1854:2
1851:e
1847:1
1844:e
1840:j
1837:e
1833:i
1830:e
1818:n
1815:e
1777:2
1774:e
1763:1
1760:e
1753:n
1750:e
1746:2
1743:e
1739:1
1736:e
1734:{
1724:F
1675:n
1672:v
1668:2
1665:v
1661:1
1658:v
1656:(
1650:n
1647:e
1643:2
1640:e
1636:1
1633:e
1631:(
1613:V
1601:n
1598:e
1594:2
1591:e
1587:1
1584:e
1568:V
1556:n
1553:v
1549:2
1546:v
1542:1
1539:v
1522:.
1499:2
1494:|
1487:n
1483:a
1478:|
1474:+
1468:+
1463:2
1458:|
1451:2
1447:a
1442:|
1438:+
1433:2
1428:|
1421:1
1417:a
1412:|
1408:=
1403:2
1393:n
1387:e
1379:n
1375:a
1371:+
1365:+
1360:2
1354:e
1346:2
1342:a
1338:+
1333:1
1327:e
1319:1
1315:a
1305:;
1302:]
1297:n
1293:a
1289:,
1283:,
1278:1
1274:a
1270:[
1262:a
1246:n
1242:e
1238:2
1235:e
1231:1
1228:e
1181:V
1149:,
1116:j
1113:i
1083:j
1080:i
1072:=
1064:j
1060:u
1056:,
1051:i
1047:u
1040::
1037:j
1034:,
1031:i
994:V
985:}
978:,
973:n
969:u
965:,
959:,
954:2
950:u
946:,
941:1
937:u
932:{
902:V
866:2
857:+
852:2
844:=
839:1
809:)
802:2
793:+
788:2
779:(
769:=
766:)
761:1
753:(
714:2
697:=
692:1
661:0
658:=
653:2
637:1
623:+
618:2
602:1
571:)
568:1
565:(
548:2
544:1
530:)
527:3
524:(
504:)
501:2
498:(
471:1
468:=
461:2
454:2
450:y
444:+
439:2
432:2
428:x
402:1
399:=
392:2
385:1
381:y
375:+
370:2
363:1
359:x
332:0
329:=
324:2
320:y
314:1
310:y
306:+
301:2
297:x
291:1
287:x
269:v
265:v
258:u
254:u
247:v
243:u
235:v
231:u
227:2
223:1
219:2
215:1
211:2
207:2
203:v
199:1
195:1
191:u
148:x
140:x
134:=
127:x
20:)
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