Today : § 1.7 : Linear Independence

Next : § 1.8 : Linear Transformations

Reminders :

My Math Lab Homework # 1 of # 2 : Due TONIGHT /

MATLAB Homework # I : Due TONIGHT 1

MATLAB Homework # 2 : Due this Friday .

Midterm 1 : Next Wed,

Jan 31,

8- 10pm .

Office hour : tomorrow ( Tue ) 9:30 - 11:00 am.

Consider the following three vectors.

x= Zzytv ← I e- span { y,

v. )! ¥

v. = ± - zy } 34+21-2 d =D

lmeaty dependent y= Est . } ," nontrivial lmewcemb

.

"

How about these three vectors :

linearly dependent.

→ Haff'f¥fsa+3b=e

a.be - 2gt3b - g =p

Definition .

A collection of vectors µ, ,u. ,...

,un] B called

linearly dependent if at least one of them

is in the span of the others.

( Yeats!It:S":#intake::e9nY '

there Is a non - trivial linear Gwb.

of th vectors= e. .

( he 4) ×, y ,

+ Xz Us + Xzy ,+ Xyby=p where at least one

of x. Tito .

¥2 bit Yz + ¥243 + ¥2934toSuppose xito

yz = f¥D y , -1¥) B- Htfjua

Theorem. Vectorsa

, uz ,...

,un are linearly independent

if and only if the vector equationX ,

U,

+

XzUz + - - - +

XnUn = O

has only the trivial solution X. =k= . . . =Xn=o .

Corollary .

The columns of a matrix A are linearly

independent iff A x.=a has only the

trivial solution to .

'⇒

kilt.nl#wftHnHtYiisttiIeiHkiyi:idie.onythetrtvfif3sotn..:lnewbrdpodntldoIogH

Columns,

estate

.int#IiitstIiIiHhYil~ non

- prootal

ivfnee Variable

foggily ) I:8nEi%n9oYnsi. Columns are

thereby dependent

What if one of the vectors is 0 ?

Always In. dependent. % a , y; d

o y , to yz to -43+83,6%15 e =p

What if there B only one vector ?

Ican Ifd ay =e fox,±o{ no'

& #e

yes,

'

fee.

What if there are only two vectors ?

X, Ytxztz =P

* o

'⇐ ¥ .v.

III ' ¥41 yes " '

e.gl?j.Ht4al

' ⇒ titty.HN km;¥gm&l ; :p :p

i$lnmpfodent .

treevarrablp

Theorem : The columns of a matrix A are linearly

Independent iff A has a pivot positionin every column

.

( otherwise,

the first non - pivotal column

is a linear combination of the precedingones

,with coefficients read off of the rref. )

Corollary : If A has more columns than rows,

its columns are linearly dependent .