Spanning sets and linear independence

Let Ji Tk E R F q EIRT linear combination of vectorssetof

column vectors VT s k with coefficients c Ckwith n real entries

Let Ig I

a Is it a linear combination of it

b Is we1Sof a we want to find c C s t c Tita I

lin.sy.sk aug matrixC Cz I

c I's iii I I's a I's I3C 1 SC 3

c a E E E1

aCz 2

RREF solutionexists 3T't 2 I

b c It a his

I's KI IIT isnot a Cmcomb of VT VT

theorem A system of linear equations with augmented matrixt

A II iff 5 is a lin combination of the columns of A

def If S it Ji VI is a setof vectors in IR

then the set of all linear combinations of it ite is called the

spay of VT Tue denoted span ie on span S

If Span S IR then S is called a spanningset for IR

E Show that 1122 span IL IE ta

Sol want to showthat forany vector f er the system c L 1 2 Ihas a solution

ausmat 231 to II I zaREF does not contain o ol o row

rfaII.io II7tzIIiI za yj msn.am

c e Tat2b solutionCz Za b

Ree span z I 1,7 is also 1122

since each of can bewritten as c L ta to 1,25

sinantIE IE

iRss ieIEI aIEJ i5IItLYSimilarly.IR spanCEi e where ETE k th place

Ex span III IIII'is II f SI's it

s t xanother description of thisplane I I e system t y is consistent

b s t 35 of z

AusMat fig IE g 11consistent if z sx o

equatoroftheplane is Z 3

Linear independence

def A set of vectors J Jn im is linearly dependent if there are

scalars 9 scm atleast one of whiseh S nonzero sat

x C I t Cmf'm 8 linear dependence relationA set of vectors that is not lin dep is calledicarly dependent

them Vectors Vi I'm in IR are lin idependent off at least one of themcan be expressed as a linearcombination of the others

1 say ate in C divideby a I Effi CqTm JEI The set E Ji m

is lie dep since 1 It 0 The to Tm T

anyvectors

two vectors it in de en dependent itf one is a multiple of theother

Ex It I ein indep III I eindep

fl III III ender L'tI in its

sac tiffs I it I I'dIg

c 35 0 co free varz 25 0 as many solutionsin part nonzerosolutions

C Cfreevariable vectors are eh dependent

Eg setting Cy I C 3 Cz 2

ji ziti Jj I lineardependence relation

The Let it Fm EIR and let A bethe mom matrix Jive Jmwiththesevectors as columns Then I Jn are linearly dependent Iffthehomey linsys with augmat Ji Fm to has a nontrivial solution

RREF

IE.ei.is's gettin29 since I I notreevariables

C Cz Czleadvars

The Any set of m vectors in IR is ln dep if m n

Ex III 131,133 ender

M 3 n 2

Remarki e for THE in 1122 or 1123 span T is a line

for Ji in IR or 112 l dp spanCT is a plane

for it Is in IR l G spanca 5 IR is thespace