linear algebra davidharbater/314-1.pdf · 2019. 1. 18. · will review both theory t computations...
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![Page 1: Linear Algebra Davidharbater/314-1.pdf · 2019. 1. 18. · will review Both theory t computations Alternative Math 312 B12has less theory more Computations 312 doesn't give Math major](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc0dc0154c8647292010d7a/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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