Math 314 Spring 2019Linear AlgebraProf David Harbater

Study of vector spacesrelated Concepts

Assumes familcity with basic

fromm Math 240 vectors in n sp esmatrices

will review

Both theory t computationsAlternative Math 312

B12 has less theory more Computations

312 doesn't give Math majorcredit

314 TA's labs 2hrslwk

Ben Foster 402,403Tu Th 6 30 830

Juan Lanfranco 404 405

Labs are active start th this wk

weekly problem sets due in lab

Three exams not cumulative in class

No final exam

See web page for more info

www.math.upeun.edu nharbateCourse 7 Math 314 Spring 19

Text Hoffman Kunze ed

MPA Library bookstore web

314 pre requisite forM h 320 1 Helpful fee 360 l

A more theoreticalthen 314also assume 240

See web pg forproblem sets examsgradinginfoGrades exams homework participateGrades posted on CanvasStudy groups for homework

Sign up pageindicate by lab

Today Vectors ScalarsVector spaces fields

Vector spacesVectors in plane

3 space

planI s

plane IR

3 space cars IRBn space 112

TEE T.iii.ieVfw lait b antbe

Cv

e 9 at ca ca

0k in

Vectors t scalars

obey alg laws

Scalars IR

both Satisfy Comm law

g assoc law

identities t 0

at a

Ia I a

Inverses a a

Atl a Oa a iyaexe fo O

a a

distrib law of overt

Summarize IR form a field

Other filescomplex s

et bi a.be R

Q1 ration.at

saTa.bmti7IIoa.bEZtF2field of 2 elements

11

90,13

toiI O

3,4 5,478,9

eeter Spaces

IRA additionScalar malt

Alg properties

Assoc Comm I ty

inverse V V

scaler mult

C EIR v EIR'sCV C IR

Scalar Vector Vector

Assoc G V C Crv

id I V V

distrib Iansc Vtu a tow

a U C V t GV

Can check by Coords

tw W TV

la al w Ch b

btw 9th Anthon

beta KatanWtv

Summarize IR is a

with Sea.la R

Their ex's of V S

over IRl 2 3 reel matrices

C c S2 Sequences of real s

A Q 93D o o o

3 Real Valued functionson Co DSinxtex

i identically 04 Polys with real coeffs

3 42 5O O p ly

V s's over other feces

Cdn n tuples of CX Sfgn rat's sheFn alts off

held IF2 3 CX matrices

11C file

Seq's of elements in F If

From basic properties of

a fete or a U s

can deduce other propertiesEx F field AEF

O A D good.ieProof 0 a Oto a

distO a to a

O a C co a

7 a to a f to.atIts IRAS

Tedd inv

LHS assoc

RH 5 0 at fo att Coram

O at 0 aed inv

O a ked id

i O a 0

V s Q V 0

Seele Oh Ved O

Pf is sane if replace

a E F by V EV

Another propertyof fields

AEF f 1 as a

Pf WT S

f 1 at a 0A s

LH s Eam at C naIn a C M a

m it idI

I te D a

dist 0 a add inv

Ow prev re alt

V S C 1 v V

Pf is sane repleaeC by V

he Vectors spaceIn side another V S

SubspaceEX spleneao CHP

Solins to f t 3 f't Zf o

cont fms

conv Sep's C sequencesupper Sr C 3 3 mis3 3 inxs

Subspace swe VV S US

with Sana t Se maltt so malt O

How to check if w is

a sub spa e

Some properties are

inherited frm V tow

assoc dat et

Still need

wtf0 EW

closed under t Sc meet

add riv's

Sats to check

W 84 closer uncle t

Why W 70 ZwewO W E W c 1 uncle r

O

Wow w c Wc 1 under C 1 W

A way to get a

Subspace of v s VTake a stat Sev

Q S t onset tarr

line Cu b of

finite men elementsof S

Let W the set of all these

W is a s b space of V

the s b speanse Spannedby S

V IRS v 3Get W Plane i gridOr W Line if

Vgrw is malt of the other

Or W O if f w o

In general for any

set S Vvs

get W CU a subspace

subspacespanned by S

Note W is non emptyclosed under t