let ji tk r f q set of linear of vectorstaylor/math20580/notes/mnev/0210.pdflet ji tk e r f q eir t...
TRANSCRIPT
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