0000001328 00000 n 993 762 272 490] /Widths[323 569 938 569 938 877 323 446 446 569 877 323 385 323 569 569 569 569 569 153 0 obj << /Linearized 1 /O 155 /H [ 1328 1855 ] /L 264253 /E 71027 /N 28 /T 261074 >> endobj xref 153 44 0000000016 00000 n 631 712 718 758 319] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 576 772 720 641 615 693 668 720 668 720 0 0 668 /BaseFont/TSMOEU+CMBX12 353 503 761 612 897 734 762 666 762 721 544 707 734 734 1006 734 734 598 272 490 This book is directed more at the former audience 0000003160 00000 n 4.1 Vector Spaces & Subspaces Math 2331 { Linear Algebra 4.1 Vector Spaces & Subspaces Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu �d$�+aX�ʯ�'���_ endobj Vector space: informal description Vector space = linear space = a set V of objects (called vectors) that can be added and scaled. << Applied Linear Algebra Vectors, Matrices, and Least Squares Stephen Boyd Department of Electrical Engineering Stanford University Lieven Vandenberghe Department of Electrical and Computer Engineering University of California, Los Angeles. 0000008612 00000 n 778 778 778 778 778 778 778 778 778 778 778 889 889 778 778 778 778 778 778 778 778 /FirstChar 33 /Type/Font /FirstChar 33 /Name/F7 278 833 750 833 417 667 667 778 778 444 444 444 611 778 778 778 778 0 0 0 0 0 0 0 ����01��B�h�� ��oA�"rv�ǔ^�q�. 4.1 Vector Spaces & Subspaces Vector SpacesSubspacesDetermining Subspaces Subspaces Vector spaces may be formed from subsets of other vectors spaces. It is used by the pure mathematician and by the mathematically trained scien-tists of all disciplines. /Widths[622 466 591 828 517 363 654 1000 1000 1000 1000 278 278 500 500 500 500 500 >> 0000003341 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 615 833 763 694 742 831 780 583 667 612 endobj << 719 595 845 545 678 762 690 1201 820 796 696 817 848 606 545 626 613 988 713 668 /Name/F5 0000010552 00000 n 272 490 272 272 490 544 435 544 435 299 490 544 272 299 517 272 816 544 490 544 517 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 676 938 875 787 750 880 813 875 813 875 /Name/F1 endobj 575 575 575 575 575 575 319 319 350 894 543 543 894 869 818 831 882 756 724 904 900 endobj /Subtype/Type1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 692 958 894 806 767 900 831 894 831 894 601 542 529 531 415 681 567 831 659 590 555 394 439 740 575 319 0 0 0 0 0 0 0 0 0 589 524 530 539 432 675 571 826 648 579 546 399 442 730 585 339 0 0 0 0 0 0 0 0 0 /Widths[718 529 692 975 612 424 747 1150 1150 1150 1150 319 319 575 575 575 575 575 /LastChar 127 /FirstChar 33 982 511 631 971 756 1142 950 837 723 869 872 693 637 800 678 1093 947 675 773 447 313 563 313 313 547 625 500 625 513 344 563 625 313 344 594 313 938 625 563 625 594 %PDF-1.3 %���� So Linear Algebra studies these linear spaces and the maps between them that are compatible with the linear structure: linear maps. 1144 875 313 563] G. NAGY – LINEAR ALGEBRA July 15, 2012 1 Overview Linear algebra is a collection of ideas involving algebraic systems of linear equations, vectors and vector spaces, and linear transformations between vector spaces. /Name/F6 0 0 772 640 566 518 444 406 438 497 469 354 576 583 603 494 438 570 517 571 437 540 446 451 469 361 572 485 716 572 490 465 322 384 636 500 278 0 0 0 0 0 0 0 0 0 0 0 0000034619 00000 n Linear Algebra Chapter 11: Vector spaces Section 1: Vector space axioms Page 3 Definition of the scalar product axioms In a vector space, the scalar product, or scalar multiplication operation, usually denoted by , must satisfy the following axioms: 6. 0000008390 00000 n /Type/Font 446 453 446 631 600 815 600 600 508 569 1139 569 569 569 0 0 0 0 0 0 0 0 0 0 0 0 2 0 obj 1. u+v = v +u, endobj /Type/Font x��=ks#��߷j��LY���5�ƦH������n �HI����$��n��h�o����Z��o޼m�juS_��޼۶����o>~���|_]7��m��7?�_���w�m[���ٻo߳__�J���HKY^ 0000003602 00000 n /FontDescriptor 35 0 R 459 444 438 625 594 813 594 594 500 563 1125 563 563 563 0 0 0 0 0 0 0 0 0 0 0 0 717 0 0 880 743 648 600 519 476 520 589 544 423 669 678 695 573 520 668 593 662 527 << 4 0 obj 0000005710 00000 n /Filter[/FlateDecode] /FontDescriptor 26 0 R /Widths[278 500 833 500 833 778 278 389 389 500 778 278 333 278 500 500 500 500 500 Important note: Throughout this lecture F is a field and V is a vector space over F. 0. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833 778 694 667 750 722 778 722 778 /BaseFont/AEWFRR+CMMIB10 trailer << /Size 197 /Info 151 0 R /Root 154 0 R /Prev 261063 /ID[<7b5792342f141717b1fdb30e9b89f253>] >> startxref 0 %%EOF 154 0 obj << /Type /Catalog /Pages 148 0 R /Metadata 152 0 R /PageLabels 146 0 R >> endobj 195 0 obj << /S 2033 /L 2251 /Filter /FlateDecode /Length 196 0 R >> stream 0000058829 00000 n Closure: The product of any scalar c with any vector u of V exists and is a unique vector of 250 459] /FirstChar 33 0000023290 00000 n /BaseFont/NWRXPS+MSBM10 These are called subspaces. 0000011383 00000 n 0000056276 00000 n The space of linear mappings from V1 to V2 is denoted L(V1,V2). /Name/F9 /FirstChar 33 /BaseFont/XHIEVD+CMBX10 /Widths[720 540 690 950 593 439 751 1139 1139 1139 1139 339 339 585 585 585 585 585 474 454 447 639 607 831 607 607 511 575 1150 575 575 575 0 0 0 0 0 0 0 0 0 0 0 0 However, it is exactly this level of abstraction that makes Linear Algebra an extremely useful tool. 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