11 Nullspaces column spaces and lineartransformations

10211912020

Nul A Ie IR s t AEsto subspace of IRRecall for A mxn matrix go A span I pig 15cRms.LI A forsoftersubspace of Rm

candescribe as Spanth Tpwith it Tvp fromtheparametricvectorsolutionof A I 8

def A l n transf T fron a v sp V into a vsp W is a rule assigning to eachvector tx in V a vector TLE in W s.tni T Itu Thi c Tat ii T cli c Thi

any ii 5

k or nullspace of T setof all I n U sit TCI7 8 E subspaceofrage of T all vectors of fern TCI in W subspaceofw

Code

iT V412 W Rm Ker T _Nul A

I AI range T 61 AFanmatrix

V functions on a b uhud have continuousderivativesw continuous functions on a b

D V 7 W Enea transf Ker D constant factors on a bTf f ranged W

411 linearlyi dependentsets.baVectors Ii Ip in a rectSp V are linearly independentthe vect.eq.ci cpJp ofc

i

has only the tri v Sol C Cp 0

Otherwise set Ji p is lin dep and withnotallweightszero is a lindeprelamong it itp

as in IR I is linindep iff f toit 3 ein dep iff E cii or ii do

I i Tip is lin dep

Thg Set ii Tp of p e vectors inV withTito is leader ifsome j g 1 is a lie comb of T Tj i

Note for FIR C cannot be cast as matrixeq AE 8 Wemustrely ondef andTkm forCinCDdependence

V IP pets 1 pacts t pzcti z.atthen pi pi ps3 is p because pz 2 z

Set Sint cost is li indep in CEO i spaceofcontinuousfunctionsonostelsince there is no scalar C sit cost c Sint for all te o D

set sent cost suet is ein der a Cfo 17 since nztstTostfVt.defLetHbeasubspaceofv.sp.VA setofvectors 13 15 Tsp in Vis a basis for It if

i B is a lie indep setCii H Span 15 Bp

let A invertible non matrix A Eat In Then columns of Ayo form a basis for IR they are lin nder ol span IR by InuMatithm

let I E 1 be alums of In

g te Ten standardbasis for IR

Ex i L E If I 11 a ii Ii a basisfor IR

Sol A ii In fo so A invertible YES

i let S ft t t t'S S basis forPn the standard basesIndeed S spans 1pm Linear independence assure G Ltc tt tCn't _OCT

then C C G o Zeropolyne

thespaneng settlea17 iii I i's17g H sina.si453

Note Js 5J tsui Qi Ca shoothat H Spentin 43Ibfadabasisforth

Soli Ca a vector Span ii Jig is c Ji t Catz to Vj in Ha vector in H is cT.ecvrtCsVJ Cces g J curses z in Spondir

5ThbBEV lander not multiples of oneanother

Spen H due to Ca

THM spanningsettheorem

Let 5 it ip3 be a spanning set for Ha if a vector VI from S is a le comb of othervectors in S then thesetformed from 5 by deleting The stillspans Hb if H f I some subset of S is a basis for H

A basis for It is i thesmallest spanningset f Hcandelete 1indepvectorsfor aspanningsetfor It Whenwe arrivetoa en hdcp subset if wedelete onemore theresultwill nolongerspanHthelargestenindepset in H

i 9 Tries

µ.li 1 elIlilIHEIa lI3 E.HEI.IDto abasis enlargement

ein depset basis in 1123 spansIR eindepdoesnotspanIRS