linear algebra davidharbater/314-1.pdf · 2019. 1. 18. · will review both theory t computations...
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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
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