linear-algebra - 2XWOLQH RI (OHPHQWDU\ 0DWUL[ $OJHEUD ,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2XWOLQH RI (OHPHQWDU\ 0DWUL[ $OJHEUD , 0DWULFHV $ $ PDWUL[ LV D UHFWDQJXODU DUUD\ RI HOHPHQWV DUUDQJHG LQ P URZV DQG Q FROXPQV RI RUGHU P[Q $ = ! " $ # # # # D D D D D D D D D Q Q P P P Q L L M M O M L $Q [Q PDWUL[ LV D URZ YHFWRU RI RUGHU Q E = > @ E E E Q L $Q P[ PDWUL[ LV D FROXPQ YHFWRU RI RUGHU P F = ! " $ # # # # F F F P M % 5XOHV DQG GHILQLWLRQV IRU PDWUL[ RSHUDWLRQV (TXDOLW\ RI WZR PDWULFHV $ % LI $ DQG % DUH RI WKH VDPH RUGHU P[Q DQG D LM E LM IRU DOO L P DQG M Q 0DWUL[ DGGLWLRQ ,I $ DQG % DUH RI WKH VDPH RUGHU P[Q WKHQ $ % & VXFK WKDW & LV DOVR RI WKH VDPH RUGHU DQG D LM E LM F LM IRU DOO L P DQG M Q 6FDODU PXOWLSOLFDWLRQ ,I LV D VFDODU WKHQ $ > D LM @ L H HDFK HOHPHQW RI $ LV PXOWLSOLHG E\ 0DWUL[ PXOWLSOLFDWLRQ ,I $ LV RI RUGHU P[Q DQG % LV RI RUGHU Q[S L H WKH QXPEHU RI URZV LQ % HTXDOV WKH QXPEHU RI FROXPQV LQ $ WKHQ $ DQG % DUH FRQIRUPDEOH IRU PDWUL[ PXOWLSOLFDWLRQ *LYHQ WZR FRQIRUPDEOH PDWULFHV $ DQG % WKH SURGXFW $% & LV GHILQHG WR EH WKH PDWUL[ RI RUGHU P[S VXFK WKDW F D E LM LN NM N Q = = L P DQG M S 7KDW LV WKH LM WK HOHPHQW RI & LV IRUPHG E\ PXOWLSO\LQJ HDFK HOHPHQW RI WKH L WK URZ RI $ WLPHV WKH FRUUHVSRQGLQJ HOHPHQW RI WKH M WK FROXPQ RI % DQG DGGLQJ DFURVV N Q = K Q HTXDOV WKH QXPEHU RI FROXPQV RI $ DQG WKH QXPEHU RI URZV RI % 1RWH WKDW %$ LV QRW GHILQHG XQOHVV P S &RPPXWDWLYH ODZV D $GGLWLRQ $ % % $ ZKHQHYHU HLWKHU RQH LV ZHOO GHILQHG E 0XOWLSOLFDWLRQ $% %$ H[FHSW LQ WKH VSHFLDO FDVH RI VTXDUH V\PPHWULF PDWULFHV $VVRFLDWLYH ODZV D $GGLWLRQ $ % & $ % & E 0XOWLSOLFDWLRQ $% & $ %& LVWULEXWLYH ODZV D 0DWUL[ PXOWLSOLFDWLRQ $ % & $% $& E 6FDODU PXOWLSOLFDWLRQ $ % $ % $ $ $ & 8QLW RU LGHQWLW\ PDWUL[ 7KH LGHQWLW\ PDWUL[ , Q LV D VTXDUH PDWUL[ RI RUGHU Q[Q ZLWK RQHV RQ WKH SULQFLSDO GLDJRQDO DQG ]HURV HOVH ZKHUH , Q = ! " $ # # # # L L M M O M L $ VFDODU PDWUL[ , Q LV D VTXDUH Q[Q PDWUL[ ZLWK D FRPPRQ VFDODU HOHPHQW RQ WKH SULQFLSDO GLDJRQDO DQG ]H URV HOVHZKHUH 1RWH $ , Q $ $ , Q $ $ GLDJRQDO PDWUL[ D LV D VTXDUH Q[Q PDWUL[ ZLWK VFDODU HOHPHQWV QRW QHFHVVDULO\ HTXDO RQ WKH SULQFLSDO GL DJRQDO DQG ]HURV HOVHZKHUH = ! " $ # # # # L L M M O M L Q 7UDQVSRVLWLRQ RI D PDWUL[ 7KH WUDQVSRVH RI $ GHQRWHG $ LV WKH PDWUL[ REWDLQHG IURP $ E\ LQWHUFKDQJLQJ URZV DQG FROXPQV $ = ! " $ # # # # D D D D D D D D D Q Q P P P Q L L M M O M L = ! " $ # # # # $ D D D D D D D D D P P Q Q P Q L L M M O M L $ V\PPHWULF PDWUL[ $ $ LV D VTXDUH Q[Q PDWUL[ VXFK WKDW D LM D ML IRU DOO L M Q ,I $ LV RI RUGHU P[Q P Q WKHQ ERWK $$ P P DQG $ $ Q Q DUH V\PPHWULF WKRXJK RI GLIIHUHQW RUGHUV $$ = ! " $ # # # # ! " $ # # # # = ! " $ # = D D D D D D D D D D D D D D D D D D D D Q Q P P P Q P P Q Q P Q LN MN N Q L L M M O M L L L M M O M L ZKHUH L M P RQ WKH IDU ULJKW KDQG VLGH = !...
View Full Document

This note was uploaded on 08/01/2008 for the course ARE 210 taught by Professor Lafrance during the Fall '07 term at University of California, Berkeley.

Page1 / 13

linear-algebra - 2XWOLQH RI (OHPHQWDU\ 0DWUL[ $OJHEUD ,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online