普通地质学3.1 Definitions of vector spaces普通地质学.pdf

Definitions of vector spacesDefinitions of vector spaces Introduction The operations of addition and scalar multiplication are used in many diverse contexts in mathematics. Regardless of the context, however, these operations usually obey the same set of algebraic rules. Thus, a general theory of mathematical systems involving addition and scalar multiplication will be applicable to many areas in mathematics. Mathematical systems of this form are called vector spaces or linear spaces. Introduction Vector space is not just a mathematical concept. The results about vector spaces have some applications. In this section we will give the definition of a vector space, show some examples and discuss some fundamental properties of vector spaces. Outline 1. Definition of vector spaceof vector space 2. Examples of vector spaces 3. Fundamental properties of vector spaces Definition of vector space Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique element x + y that is also in Vin V, and with each element x in V and each scalar , and with each element x in V and each scalar αα, , we can associate a unique element αx in V. Namely, linear combination αx + β y in V. The set V, together with the operations of addition and scalar multiplication, is said to form a vector space if the following axioms are satisfied: A1. x + y = y + x for any x and y in V. A2. (x + y) + z = x + (y + z) for any x, y, and z in V. A3. There exists an element 0 in V such that x + 0 = x for each x ∈V. A4. For each x ∈V, there exists an element −x in V such that x + (−x) = 0. A5. A5. αα(x + y) = (x + y) = ααx + x + ααy for each scalar y for each scalar αα and any x and any x and y in V. A6. (α + β )x = αx + β x for any scalars α and β and any x ∈V. A7. (αβ)x = α(β x) for any