Vector Spaces

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

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Vector Spaces

  1. 1. Vector Spaces Chapter 18
  2. 2. What is a vector space? <ul><li>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: </li></ul>
  3. 3. Proposition 18.1 <ul><li>Let V be a vector space over F. Then each of the following statements is true. </li></ul>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. 4. Example <ul><li>Show that the n-tuples are a vector space over R. </li></ul>
  5. 5. Example <ul><li>What about the field ? Is it a vector space? </li></ul>
  6. 6. Subspaces <ul><li>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. </li></ul>
  7. 7. Example <ul><li>Show W be a subspace of where </li></ul>
  8. 8. Terminology <ul><li>Linear Combination </li></ul><ul><ul><li>Any vector w in V of the form </li></ul></ul><ul><ul><li>Is the linear combination of the vectors </li></ul></ul><ul><li>Spanning Set </li></ul><ul><ul><li>The set of vectors obtained from all possible linear combinations of </li></ul></ul>
  9. 9. <ul><li>Proposition 18.2 </li></ul><ul><ul><li>Let be vectors in a vector space V. Then the span of S is a subspace of V. </li></ul></ul>
  10. 10. Linear Independence <ul><li>Let be a set of vectors in a vector space V. If there exists scalars </li></ul><ul><li> such that not all of the </li></ul><ul><li>are zero and </li></ul><ul><li>then S is linearly dependent. </li></ul><ul><li>If all the scalars are zero, then S is linearly independent. </li></ul>
  11. 11. <ul><li>Proposition 18.4 </li></ul><ul><ul><li>A set of vectors in a vector space V is linearly dependent iff one of the is a linear combination of the rest. </li></ul></ul><ul><li>Proposition 18.5 </li></ul><ul><ul><li>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. </li></ul></ul>
  12. 12. Basis <ul><li>A set of vectors in a vector space V is called a basis for V if is a linearly independent set that spans V. </li></ul>
  13. 13. Example <ul><li>Find a basis for </li></ul>
  14. 14. Example <ul><li>Find a basis for </li></ul>

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