Chapter 2A Reading Notes

title: Review
F denotes R or C
V denotes a [[file:KBe20math530refVectorSpace.org][KBe20math530refVectorSpace]] over F 

• lin alg does not focus on arbitrary vector spaces
• it focuses on finite-dimensional vector spaces!

title: learning objectives for the chapter
- span //covered in section
- linear independence //covered in section
- bases 
- dimension

• notation:
  • lists of vectors:
    • (2,1,4),(3,2,5)
      • list len 2 of vectors in R³
    • n-tuples without surrounding parens
• linear combination
  • a linear combination of x and y would be any expression of the form ax + by, where a and b are constants ~wiki
  • multiply each element in a list of vectors by an element in F
  • and then add them up!
  • any relation between the element scalar and what's being multiplied? can the scalars repeat? #question
    • yes, yes they can.
• span
  • the set of all linear combos of a list of vectors
    • denoted: span(v1,…,vm)
  • span of empty list is {0}
  • aka. linear span
• KBxSpansLinAlg

the span of a list of vectors in V is the smallest subspace of V containing all the vectors in the list

```ad-question
but don't you get out a single vector at the end..? because you add them? #question no! because it's the *set* of all linear combos

• *finite-dimensional vector space
  • a vector space is called finite-dimensional if some list of vectors in it spaces the space
    • spans the space..?
    • ????
• linear independence
  • a list of vecors in V where the only choise of a1 … am in F that makes a1v1 + … + amvm = 0 is a1 = … = am = 0
  • uniqe way to get 0?
• lineary dependant
  • opposite, can get to 0 with non-zero scalars
• KBxLinearIndependence

#review the end here #todo some exercises