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Vector Spaces

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Franklin College Mathematics and Computing Department
Franklin College Mathematics and Computing Department

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

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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

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Vector Spaces

  • 2. 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:
  • 3. 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.
  • 4. Example Show that the n-tuples are a vector space over R.
  • 5. Example What about the field ? Is it a vector space?
  • 6. 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.
  • 7. Example Show W be a subspace of where
  • 8. 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
  • 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 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.
  • 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 a basis for
  • 14. Example Find a basis for