TR3.5

Chapter 1.B

Huxley Marvit 2021-09-27 Mon 12:00

#flo #ref #hw

1 def of a vector space

  • Props of addition and scalar multiplication in FN
    • +: comutative, associative, identiy
      • every element has an additive inverse
    • *: associative, identity
    • addition and scalar multiplication, connected by distributive props
  • let V be a set with an addition and scalar multiplication that satisfy the props,
**addition, scalar multiplication**
- addition: assigns an element u+v in V to each pair of elements u, v in V
- scalar multiplication: lv with l in f and v in V

**vector space**
is V with addition and SCMUL with:

- commutativitity
- associativity
- additive idenitity
- additive inverse
- multiplicative identity
- distibutive properties
  • no multiplicative inverse?
    • is this how you solve the 0 issue?
  • vec, point
    • elements of vec space are called vecs or points
  • simplest vec space: \(\{0\}\)
  • finfin is the set of all seqencues of elements of F
    • additive identity: seqnece of all zeros
  • vector space can include a set of functions? not quite..
    • let S be a set, and FS be the set of functions from S to F
    • what?? #review
  • let S be the interval [0,1] and F=R
    • R^[\0, \1] is the set of real valued function on the interval [0,1]
    • ??
  • FN -> F1,2,…,n
  • Finfin -> F1,2,…
  • vector spaces need unique additive inverse
    • 0'=0'+0=0+0'=0
      • nicer than my proof
  • unique additive inverse
    • w=w+0=w+(v+w')=(w+v)=(w+v)+w'=0+w'=w'
V denotes a vector space over F

1. no multiplicative inverse required?
2. what does the set of functions from S to F mean?

1.1 exercises

  1. prove that -(-v) = v
    1. -(-v) = -1(-1v) = (-1*(-1))v = 1v = v
  2. ab = 0, prove that a or b = 0
    1. a=0/v = 0, v=0/a = 0
  3. empty set is not a vector space, it fails to satisfy only of the reqs. which one?
    1. no additive idenity
      1. "there exists an element 0 in v" no there doesn't.

homework: KBxSolvingSystems