lect136_14_w09

# lect136_14_w09 - Wednesday February 4 Lecture 14 Abstract...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

lect136_14_w09 - Wednesday February 4 Lecture 14 Abstract...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online