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Unformatted text preview: Chapter 5 General Vector Spaces 1 Introduction Let A be an m × n matrix and let b ∈ R m . Then we have a linear system Ax = b (1) with variables x = ( x 1 ,...,x n ) T ∈ R n . So far in this course we have developed methods to analyze and solve such systems. In the remainder of the course we develop methods to reduce more general types of problems to linear systems of this form. This will allow us to use our knowledge of linear systems to solve a much broader and more interesting class of mathematical problems. Let V and W be sets and let T : V → W be a function (“transformation”). In this setup we can consider the problem of solving T ( f ) = g (2) where g is a given element of W and f is a variable element of V which we wish to solve for. We are going to find general conditions on V , W and T which allow (2) to be reduced to a linear system of the form (1), which we can then solve using our knowledge of linear systems. The specific conditions are that V and W should be general vector spaces (with finite dimensions), and T should be a linear transformation . We discuss general vector spaces in this chapter and linear transformations in the following chapter 6. 2 General Real Vector Spaces: Definition and Examples 2.1 Motivation The Euclidean n-space R n is a set on which addition and scalar multiplication are defined: ( u 1 ,...,u n ) + ( v 1 ,...,v n ) = ( u 1 + v 1 ,...,u n + v n ) c ( u 1 ,...,u n ) = ( cu 1 ,...,cu n ) Addition and scalar multiplication on R n satisfy certain properties: associative law for vector addition, distributive laws, there is a zero vector, multiplying a vector with the scalar 1 has no effect etc. Aside from R n , we can also consider other sets on which addition and scalar multiplication are defined and have the same properties. Such sets are call “real vector spaces”. 2.2 Definition A real vector space is an nonempty set V on which addition and scalar multiplication are defined (this means that for any u,v ∈ V and c ∈ R it must be defined what u + v and cu are) such that the following conditions are satisfied for all u,w,v ∈ V and all c,d ∈ R . 1. u,w ∈ V ⇒ u + w ∈ V (closed under addition) 2. u ∈ V , c ∈ R ⇒ cu ∈ V (closed under scalar multiplication) 3. There is an element 0 ∈ V such that u + 0 = 0 + u = u for all u ∈ V (existence of zero vector) 4. For every u ∈ V there is an element − u ∈ V with u + ( − u ) = 0 (existence of negative vector) 5. ( u + v ) + w = u + ( v + w ) (associative law for addition) 6. u + v = v + u (commutative law for addition) 7. c ( u + v ) = cu + cv (distributive law for vector addition) 8. ( c + d ) u = cu + du (distributive law for addition of scalars) 9. c ( du ) = ( cd ) u (associative law for scalar multiplication) 10. 1 u = u (multiplication with scalar 1 has no effect) In the following, we will often simply refer to real vector spaces as “vector spaces”....
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This note was uploaded on 01/24/2011 for the course SPMS MAS 114 taught by Professor Profdavidh.adams during the Spring '10 term at Nanyang Technological University.
- Spring '10