Today : § 4.4 : Coordinates

& § 3.1-3.2 : Determinants

Next : §3.2-3.3: Determinants of Volume

Reminders

My Math Lab Homework # 5 : Due Tuesday by Hispm

MATLAB Assignment # 4.Due February23

by 11:59pm

Midterm # 2 : Feb 28,

8- 10pm

Reminder : If V is a vector Space and B :{ b., td,

...

,by) is a

basis far V,

the coordinate vector [ v.bps of any vector yet

is the column of coefficients in the unique expansion of v. in

the basis B :

minions . # Is" ¥"Yes;Ife → wasted

Theorem If B= { b. b.↳ .

,bn } is a basis for V

,then

the function T : V → Rn : TH ) = WBis a one-to-one linear transformation from V onto Rn

.

Such a linear transformation B called an isomorphism .

earthly rrett:L Hit# is beaten.

i. dim(GkAD=2 b.bz

÷anklA)

Hkitty.tn#Ht1H:k&I.l/fk=H/3ghjlNT:bnsbn-s.ei.....e.elR "

↳. |fqftoPy|

The point of isomorphism is : they preserve all linear properties .

Theorem : Let{ by he,

...

,big :B be a basis for V

.

Then * { a, ...

, b) TV ane linearly independent in V

iff LIB,

...

,[ Hops are linearly independent In IN

.

*" "

spanV

" "

span IN.

Eg .

Show that { 1

,

x - i,

( x . 15 } is a basis forPzelatbxtcx'

}P

Has =p;)https://HDYBHBai

, x. xD\fema

k

H. D'

. ii. 2×+1}"

/ ! ftp. : Yes,ban .

§ 3.1 Determinants

If A is a 2×2 matrix A = ( 9bd| ,when doing new

reduction,

we find that the columns are both pivotal

unless ad - bc =o.

-

This is the determinant of A,

denoted det A

= IAI.

Theorem : A is invertible iff det A ¥0

in which case At = adtbefdeiab )

For any nxn matrix A,

there is a quantity det A

whrh

* is a polynomial function of the entries of A

* determines whether A is invertible.

The 2×2 case has a simple formula ; for largermatrices

,it is more complicated .

The generalization of the 2×2 case B the

cofactor expansion .

⇐ HE!l Kitt on "HHEIYFAE¥6.

Cofactor Expansion

A=µ"uaahu÷aayw| detltjajicjitajicjzt. - tajncjw forany new

in,

jin.iann 9 Jcljtazjczjt ' ' ' tanjcnj orolumnj .

E -9 .

(II?fdebA= 141+542+043

At Fo=|µ ,)kH+sc.nl#+oankH( ty¥# = 114 .o - EDHD + 4) ( o ) + ol A

= -2.

HyffdetA-o.xttytoll@to.x'

= @)H)( . D= -2.

9' ⇒

attemptHatfield

0024- 1

÷.: =3

-4

!Ez÷t=n. total#

+ - t =3 -2ft)H)@

ftp.F# ¥: ¥

Theorem If A is triangular detA is the

product of the diagonal entries.