Vectors_Tensors_02_Vector_Space

Vectors_Tensors_02_Vector_Space - Section 1.2 1.2 Vector...

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

View Full Document Right Arrow Icon
Section 1.2 Solid Mechanics Part III Kelly 10 1.2 Vector Spaces The notion of the vector presented in the previous section is here re-cast in a more formal and abstract way. This might seem at first to be unnecessarily complicating matters, but this approach turns out to be helpful in unifying and bringing clarity to much of the theory which follows. Some background theory which complements this material is given in Appendix A to this Chapter, §1.A. 1.2.1 The Vector Space The vectors introduced in the previous section obey certain rules, those listed in §1.1.3. It turns out that many other mathematical objects obey the same list of rules. For that reason, the mathematical structure defined by these rules is given a special name, the linear space or vector space . First, a set is any well-defined list, collection, or class of objects, which could be finite or infinite. An example of a set might be { } 3 | = x x B (1.2.1) which reads “ B is the set of objects x such that x satisfies the property 3 x ”. Members of a set are referred to as elements . Consider now the field 1 of real numbers R . The elements of R are referred to as scalars . Let V be a non-empty set of elements K , , , c b a with rules of addition and scalar multiplication , that is there is a sum V + b a for any V b a , and a product V a α for any V a , R . Then V is called a (real ) 2 vector space over R if the following eight axioms hold: 1. associative law for addition : for any V c b a , , , one has ) ( ) ( c b a c b a + + = + + 2. zero element : there exists an element V o , called the zero element, such that a a o o a = + = + for every V a 3. negative (or inverse ): for each V a there exists an element V a , called the negative of a , such that 0 ) ( ) ( = + = + a a a a 4. commutative law for addition : for any V b a , , one has a b b a + = + 5. distributive law, over addition of elements of V : for any V b a , and scalar R , b a b a + = + ) ( 6. distributive law, over addition of scalars : for any V a and scalars R β , , a a a + = + ) ( 1 A field is another mathematical structure (see Appendix A to this Chapter, §1.A). For example, the set of complex numbers is a field. In what follows, the only field which will be used is the familiar set of real numbers with the usual operations of addition and multiplication.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

Page1 / 5

Vectors_Tensors_02_Vector_Space - Section 1.2 1.2 Vector...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online