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Lecturenotes14-16February2011

Lecturenotes14-16February2011 - Chapter 4 General Vector...

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Chapter 4, General Vector Spaces Section 4.1, Real Vector Spaces In this chapter we will call objects that satisfy a set of axioms as vectors. This can be thought as generalizing the idea of vectors to a class of objects. Vector space axioms: Definition: Let V be an arbitrary nonempty set of objects on which two operations are defined on which two operations are defined addition, and multiplication by scalars. By addition we mean a rule for associating with each pair of objects u and v in V an object u+v, called the sum of u and v; By scalar multiplication we mean a rule for associating with each scalar k and each object u in V an object ku, called scalar multiplication of u by k. If the following axioms are satisfied by all objects u,v,w in V and all scalars k and m, then we call V a vector space and we call the objects in V vectors. 1. If u and v are vectors in V, then u+v is in V. 2. u+v=v+u 3. u+(v+w) = (u+v)+w 4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V. 5. For each u in V, there is an object -u in V, called a negative of u such that u +( - u ) = ( - u ) + u = 0 6. If k is any scalar and u is any object in V, then ku is in V. 7. k(u+v)=ku+kv 8. (k+m)u=ku+mu 9. k(mu)=(km)u 10. 1u=u Note: When the scalars are real numbers, V is called a real vector space. When the scalars are complex numbers, V is called a complex vector space. Addition and scalar multiplication do not need to be the operations we defined before. 1
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Examples: 1. Zero vector space: V = { 0 } , 0+0=0, k0=0 2. The set of all n × m matrices with real entries, with addition= matrix addition , and scalar multiplication= scalar matrix multiplication forms a real vector space. (a) If A and B are n × m matrices then A+B is also a n × m matrix (b) A+B=B+A (c) A+(B+C)=(A+B)+C (d) There is zero matrix 0, such that A+0=0+A=A (e) For each matrix A we have -A such that A+(-A)=0 (f) If k is any real scalar, then kA is in V (g) k(A+B)=kA+kB (h) (k+m)A=kA+mA (i) k(mA)=(km)A (j) 1A=A, here 1 is the scalar 1. These properties hold by Theorem 1.4.1 and Theorem 1.4.2. 3. Every plane through the origin is a vector space: Here our set is the set of points of the plane, addition and scalar multiplications are the usual ones. Then we have 0 in the plane, If ax + by + cz = 0 is the plane equation, then for any u = ( u 1 , u 2 , u 3 ) , v = ( v 1 , v 2 , v 3 ) on the plane, then u+v will be on the plane as well. Other properties can easily be checked.
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