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# Vector Spaces

## on Dec 17, 2009

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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 SpacesPresentation 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