Post on 04-Feb-2021
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
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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
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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
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Another propertyof fields
AEF f 1 as a
Pf WT S
f 1 at a 0A s
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m it idI
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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