Math 314 Spring 2019Linear AlgebraProf David Harbater

Study of vector spacesrelated Concepts

Assumes familcity with basicfromm Math 240 vectors in

n sp esmatrices

will review

Both theory t computationsAlternative Math 312B12 has less theory more Computations

312 doesn't give Math majorcredit

314 TA's labs 2hrslwkBen Foster 402,403Tu Th 6 30 830Juan Lanfranco 404 405

Labs are active start th this wk

weekly problem sets duein lab

Three exams not cumulative in classNo final exam

See web page for more info

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

Text Hoffman Kunze edMPA 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 homeworkSign up page

indicate 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 antbeCv

e 9 at ca ca

0k in

Vectors t scalarsobey alg laws

Scalars IR

both Satisfy Comm lawg assoc law

identities t 0at a

Ia I a

Inverses a aAtl a Oa a iyaexe fo O

a a

distrib law of overt

Summarize IR form a fieldOther files

complex set bi a.be R

Q1 ration.at

saTa.bmti7IIoa.bEZtF2field of 2 elements

11

90,13

toiI O

3,4 5,478,9eeter 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 Coordstw W TV

la al w Ch b

btw 9th Anthonbeta KatanWtv

Summarize IR is awith 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 ofa fete or a U scan deduce other propertiesEx F field AEF

O A D good.ieProof 0 a Oto adist

O a to a

O a C co a

7 a to a f to.atIts IRAS

Tedd invLHS assoc

RH 5 0 at fo att CoramO at 0

aed inv

O a ked id

i O a 0

V s Q V 0Seele O

h Ved O

Pf is sane if replacea 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 ocont fms

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

Subspace swe VV S USwith Sana t Se maltt so malt O

How to check if w isa sub spa e

Some properties areinherited frm V tow

assoc dat et

Still needwtf0 EW

closed under t Sc meetadd riv's

Sats to checkW 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 aSubspace of v s V

Take a stat Sev

Q S t onset tarr

line Cu b offinite men elementsof S

Let W the set of all these

W is a s b space of Vthe s b speanse Spanned

by 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 anyset S V

vs

get W CU a subspacesubspacespanned by S

Note W is non emptyclosed under t