Null Spaces, Column. Transformations. Remarks. Definition ( ) ( ) Null A is the set of all x's in R that go to 0 in R 3
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1 Null Spaces Colu Spaces ad Liear rasforatios earks Whe we study fuctios f ( ) we fid that a uderstadig of the doai D f itervals of - values ad the rage f itervals of y - values are ecessary to descrie the fuctio. With trasforatio fro to we ust first fid the associated atri ad its traspose. We the ust fid the the vector suspaces which helps us descrie ad uderstad the trasforatio. We will eaie four vector suspaces two i ad two i y eaiig the atrices ad. Defiitio he ull space of a atri writte Nul or is the set of all solutios to the hoogeeous equatio { : ad } Null is the set of all 's i that go to i
2 heore he ull space of a atri is a suspace of. Equivaletly the set of all solutios to a syste of hoogeeous liear equatios i ukows is a suspace of { : } heore Proof: is a suset of sice has colus. We ust verify properties of the defiitio of a suspace. a) Sice is the trivial solutio to the hoogeeous equatio ) Suppose uv the u ad v We the have u+ v u+ v + Hece u+ v heore u N( ) ( u ) ( u ) hece cu c) Suppose ad c is a scalar the c c c Sice properties p a ad c hold is a suspace of eark: Solvig yields a eplicit descriptio of Nul
3 Eaple 9 he augeted atri correspodig to he N( ) Spa { uvw } u v w 7 earks. Spaig set of foud usig the ethod i the last eaple is autoatically liearly idepedet. { } N ( ). If the the uer of vectors i the spaig set for equals the uer of free variales i 8 Defiitio Col is the colu space of a atri Col is the set of all liear coiatios of the colus of Col is i the codoai ad is the rage of so we ca deote it [ a a ] Col Spa { a a } If the 9
4 heore he colu space of a atri is a suspace of { : for soe } Suary { : } { : for soe } Doai of Codoai of Eaple his is a trasforatio fro to Doai i Codoai i Our solutios set is the plae + y+ z We have oe pivot
5 Eaple Usig the pivot colu the asis of the rage is o otai solutio to ( ) yz y is free z is free I paraetric for y z + + Eaple z y + y+ z y age is y ll s go to the lie y s o the plae + y + z go to () Null set is + y+ z his is the plae through the origi that is parallel to + y+ z Eaple + y+ z + y+ z + y+ z
6 Eaple o otai the solutio to y y is free z is free I paraetric for y z + We ca write the asis for the Null set N( ) Eaple We try vectors i the plae + y + z We see i each case the solutio is () 7 We try vectors ot i the plae + y + z 8 We see i each case the solutio satisfies y Eaple 8
7 Choose vectors i plae + y+ z Eaple We see i each case the solutio is () 9 Eaple Let us ow cosider We have oe pivot he hoogeous he ( ) solutio is + y or y y is free is the lie fro origi through he Eaple We will copare ad N( ) ( ) y y y 7
8 Eaple We will copare ad ( ) N( ) is the plae + y+ z ( ) is the lie fro origi through ( ) Sice the oral vector to the plae a + y + cz d is ( ) a c We see that ad are orthogoal Eaple eview We row reduced to fid () ad N() ( ) N( ) We row reduced to fid ( ) ad N( ) Oserve that () ca e see i ref Oserve that ( ) ca e see i ref earks N( ) called the left ull space of is the set of all vectors such that. It is the sae as the ull space of the traspose of. he left ull space is the orthogoal copleet to () the colu space of. { : } he colu space row space ull space ad left ull space are soeties referred to as the four fudaetal suspaces. 8
9 earks Space Syol I Notes Colu () Fid Col space of or row space of Null N() Solve ow ( ) Fid Col space of or row space of Left Null N( ) Solve or () ad N( ) are orthogoal spaces i ( ) ad N() are orthogoal spaces i earks he suspaces ay e descried usig spaig vectors as log as these vectors spa the sae space. I this case we fid ay vector fro oe set is liearly depedet o the vectors i the other set. Eaple 7 9 ( ) spa 7 N ( ) spa free 7 9 ( ) spa 9
10 Eaple ( ) spa ( ) spa N( ) spa spa free Eaple ( ) spa 7 N ( ) is plae is lie orthogoal to plae ( ) spa Eaple spa is plae N( ) spa is lie orthogoal to plae ( )
11 Eaple Solutio i paraetric vector for is spa Eaple is the lie traslated parallel to 7 8 a) Fid all solutios to 9 Epress aswer i paraetric for Practice ) Fid a asis for the colu space c) Fid a asis for the ull space d) Fid a asis for the row space e) Fid a asis for the left ull space [ ] 7 hree pivot colus
12 Practice is free is free o fid asis for colu space we ( ) 8 use pivot colus o fid a asis for the ull space we solve Practice. Our paraetric result 7 is free is free 7 + Our asis for 7 the ull space N( ) is the Practice hree pivot colus which we use to fid ( ) 7 8
13 Practice free 7 earks We ca descrie a vector space with differet sets of vectors; our solutio is ot uique Each set for the sae space will have the sae uer of vectors; so i a sese they are the sae size d the vectors i each set will have the sae uer of eleets; they are the sae size lso if we copare two sets descriig the sae vector space we will discover that ay vector i oe set will e a liear coiatio of the vectors i the other set 8
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