Vector Spaces
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Vector Spaces
Equations are omitted for technical reasons - download the original pdf
Consider the two vectors [Equation goes here - download the original to see it.]
Vector addition has the following properties: (A1) Addition is closed [Equation goes here - download the original to see it.] (A2) There is a zero vector. [Equation goes here - download the original to see it.] (A3) There is an inverse vector. [Equation goes here - download the original to see it.] (A4) Vector addition is associative. [Equation goes here - download the original to see it.] (A5) Vector addition is commutative. Scalar multiplication has the following properties. (B1) Scalar multiplication is closed. [Equation goes here - download the original to see it.] (B2) [Equation goes here - download the original to see it.] (B3) [Equation goes here - download the original to see it.] (B4) Scalar multiplication is distributive over vector addition that is[Equation goes here - download the original to see it.] Exercise: Prove each of the properties of the two-dimensional vector space. [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] Any abstract structure that satisfies the axioms (properties) (A1) to (A5), (B1) to (B4) will be called a vector space. The point of this is that there are many structures that form vector spaces. For example, the set of all 3 dimensional vectors is a vector space; that is vectors of the form [Equation goes here - download the original to see it.] form a vector space. There are other important examples...
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Contents of
Vector Spaces
1 Vector Spaces
2 Linear polynomials
3 Complex numbers
4 Quadratic polynomials
5 Continuous Functions
6 Infinite sequences
7 Vector space axioms
8 Properties of vector spaces
9 Subspaces of vector spaces
10 Basis and dimension
11 Span
12 Theorem (Vector spaces)
13 Vectors, dimension and bases
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