Downloaded 447 times
Vector Spaces
The document provides an overview of vector spaces, defining key concepts such as scalar product, subspaces, linear combinations, and linear independence. It includes propositions related to these concepts and practical examples illustrating vector spaces over a field. Specific criteria for identifying bases and linear dependence among sets of vectors are also discussed.
More Related Content
Vector Spaces
- 1.
- 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