lect136_14_w09

lect136_14_w09 - Wednesday February 4 - Lecture 14:...

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Wednesday February 4 Lecture 14: Abstract vectors spaces and subspaces (Refers to section 4.1) Expectations: 1. Define abstract vector space . 2. Define closure under an operation. 3. Prove: a ) 0 · v = 0 , for all v in V , b ) α · 0 = 0 for all α in R . c ) ( 1)· v = v , for all v in V. 4. Prove: If α in R and v in V are such that α v = 0 . Then either α = 0 or v = 0 . 5. Recognize R n , the set of all m by n matrices, M n,m , and all functions f on [0,1] as examples of vector spaces. 14.1 Definitions An abstract vector space , ( V , +, scalar multiplication), is a set V along with two operations + and scalar multiplication that satisfy all the axioms in the list Vector space Axioms . 14.1.1 Remark – In the above definition we use the real numbers as scalars. This is why we often specify “vector space over the reals ”. The means that we only consider real numbers as scalars. We sometimes speak of a “vector space over the complex numbers ” as a way of saying that we accept complex numbers as scalars. 14.1.2
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lect136_14_w09 - Wednesday February 4 - Lecture 14:...

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