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# 4.1 - MATH1850U/2050U Chapter 4 1 GENERAL VECTOR SPACES...

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MATH1850U/2050U: Chapter 4 1 GENERAL VECTOR SPACES Recall: In Chapter 3, we saw n -space or R n . All together, the following 3 things make up n -space: 1. The objects 2. Rule for addition : a rule for associating with each pair of objects u and v an object v u , called the sum of u and v 3. Rule for scalar multiplication: a rule for associating with each scalar k and each object u in V an object u k , called the scalar multiple of u by k Real Vector Spaces (4.1; pg. 171) Definition: Let V be an arbitrary nonempty set of objects on which two operations are defined, addition and multiplication by scalars (numbers). If the following axioms are satisfied by all objects u , v , w in V and all scalars k and l , then we call V a vector space and we call the objects in V vectors . 1. If u and v are objects in V , then v u is in V . 2. u v v u 3. w v u w v u ) ( ) ( 4. There is an object 0 in V called the zero vector for V , such that u u 0 0 u 5. For each

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4.1 - MATH1850U/2050U Chapter 4 1 GENERAL VECTOR SPACES...

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