Today : § 4.2-4.3 : Bases of 6km,

NUKA)

& § 4.5-4.6 : Dimension A Rank

Next : § 4.4 : Coordinates

Homework :

TuesdayMymathlab Homework # 4 : Due TONIGHT by #8pm

My Math Lab Homework # 5 : Due Feb 20 by Hispm

A basis of a subspace H is a collection of vectors

{ 4, Va . .

.

,v.a } < It

that 4) spans It,

and

(2) B linearly independent .

Subspaces are frequently presented as the span of

some collection of vectors ; but those vectors may not

be independent .

How do we find a basis ?

It =

span { / to ) , IG ) , # |If * H

,then

,bosom . xnxx.IR

g- X, yt Xzt + BI

y - t = I= x

, ytxzvt X, ( it - 1)

= span { is.

v. }"nndeewpbspanmyftx,

* 3) is + we B) ± e- span { ⇒set,

i. basis.

Theorem : If H= span { a,

... ,b},

then some collection

of these vectors Is a basis for H.

y A B H I

Eg .

Hsspanlfff:#.

ftp.H.fi#=6kAa=HfaIsfYdmefll:?fI3zj

g , G 93 94 95g b I A ↳{ ↳ b }

a., = -291v. z= -24 four a betq↳ = - g , +293v. 4=-4+24 farCol (A)↳ = 3g , -43* = 3g -2×3

Theorem : The pivotal columns of A form a basis

for Col (A).

( The coefficients expressing

the non - pivotal columns in terms of this

basis can be read off of rref (A).)

' ⇒

HIM→ It.§A → HIM 1 KTEA fma

2 1 bass for Col A).

Caution : The pivotal columns of A,

net rref(A) ,

form a basis for 611A ). Typically

61 1 A) f- 61 ( rref (A) ).

How about Null A) ? How de I find a basis ?

← HE,

Is fedmet to?I?3:)

naff'x¥gff¥IE¥Y¥5fxf #*

ftp.ftxswgfspan.a.bg

lnedly independent.

Theorem : This )procedure for identifying dullA)

always produces a basis for it.

A given subspace always has zillions of bases.

But

there B one thing in common among all bases.

Theorem : Any two bases of a vector space have the

same number of vectors.

Pf. Suppose { a, a } is a bass for a subspace It .

of { g. k,

I }" µ "

So y ,= a

, y tb, yz , g. = azy , tbzyr , D= azyitbzuz

too onsrffr( of;ayof;) §.

the B some nontrivialf¥f=±AI =p

AI = / 91×1 +92×2+93×3biy+bzxtb}x}|=Tx. gtxzttxzv ,

= ×, ( aiui + b

, b) + Xz ( az ↳tKYDT 11319 } bit by .uD

= ( ×, G

, +192 t BAD 91 + ( x,

b,

+ xzbzt Bbdyz = e .

Definition. If V is a vectorspace ,

its dimension

Is the number of vectors in any basis.

Eg.

V= span # ' H' #, ,

We saw that the matrix A =fatty) has 2 pivotal

columns.

V= 61 (A),

so

21

dim ( D= 2

Eg .

B={ polynomials of degrees 3) = { a×3+bx2tcx+d }

Eg .

D= { polynomials }=

span { ×3×?×,

B

= Spas{ 1. x. x3x3,

... ,xH,

... ] DMB -4; dmP= es.