Vector Spaces

Vector Spaces

• 1.
  Vector Spaces Chapter18
• 2.
  What is avector space? A vector space V over a field F is an abelian group with scalar product, defined for all in F and all v in V satisfying the following axioms:
• 3.
  Proposition 18.1 LetV be a vector space over F. Then each of the following statements is true. For all v in V. For all in F. Then either =0 or v=0 For all v in V. For all in F and all v in V.
• 4.
  Example Show thatthe n-tuples are a vector space over R.
• 5.
  Example What aboutthe field ? Is it a vector space?
• 6.
  Subspaces Let Vbe a vector space over a field F, and W be a subset of V. Then W is a subspace of V if it is closed under vector addition and scalar multiplication.
• 7.
  Example Show Wbe a subspace of where
• 8.
  Terminology Linear CombinationAny vector w in V of the form Is the linear combination of the vectors Spanning Set The set of vectors obtained from all possible linear combinations of
• 9.
  Proposition 18.2 Let be vectors in a vector space V. Then the span of S is a subspace of V.
• 10.
  Linear Independence Let be a set of vectors in a vector space V. If there exists scalars such that not all of the are zero and then S is linearly dependent. If all the scalars are zero, then S is linearly independent.
• 11.
  Proposition 18.4 Aset of vectors in a vector space V is linearly dependent iff one of the is a linear combination of the rest. Proposition 18.5 Suppose that a vector space V is spanned by n vectors. If m>n, then any set of m vectors in V must be linearly dependent.
• 12.
  Basis A set of vectors in a vector space V is called a basis for V if is a linearly independent set that spans V.
• 13.
  Example Find abasis for
• 14.
  Example Find abasis for