Vector Spaces
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Carrie Trommater presentation on Vector Spaces for MAT 361 (Modern Algebra), Franklin College, Fall 2009.

Carrie Trommater presentation on Vector Spaces for MAT 361 (Modern Algebra), Franklin College, Fall 2009.

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Vector Spaces Vector Spaces Presentation Transcript

  • Vector Spaces Chapter 18
  • What is a vector 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:
  • Proposition 18.1
    • Let V 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.
  • Example
    • Show that the n-tuples are a vector space over R.
  • Example
    • What about the field ? Is it a vector space?
  • Subspaces
    • Let V be 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.
  • Example
    • Show W be a subspace of where
  • Terminology
    • Linear Combination
      • Any 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
    • Proposition 18.2
      • Let be vectors in a vector space V. Then the span of S is a subspace of V.
  • 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.
    • Proposition 18.4
      • A set 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.
  • 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.
  • Example
    • Find a basis for
  • Example
    • Find a basis for