Vector Spaces
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Vector space axioms
Thus a vector space is any set of elements V (called vectors) on which we can define operations of vector addition and scalar multiplication that satisfy the following axioms.A1) CLOSURE under vector addition If[Equation goes here - download the original to see it.]then[Equation goes here - download the original to see it.] (A2) IDENTITY There exists a zero vector [Equation goes here - download the original to see it.] such that [Equation goes here - download the original to see it.] for all [Equation goes here - download the original to see it.] (A3) INVERSES for all[Equation goes here - download the original to see it.]there exists a vector [Equation goes here - download the original to see it.]such that [Equation goes here - download the original to see it.] (A4) ASSOCIATIVITY [Equation goes here - download the original to see it.] for all[Equation goes here - download the original to see it.] (A5) COMMUTATIVITY [Equation goes here - download the original to see it.] for all[Equation goes here - download the original to see it.] (B1) CLOSURE under scalar multiplication If[Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.]then[Equation goes here - download the original to see it.] (B2)IDENTITY [Equation goes here - download the original to see it.] for all [Equation goes here - download the original to see it.] (B3) ASSOCIATIVITY [Equation goes here - download the original to see it.] for all[Equation goes here - download the original to see it.] (B4) DISTRIBUTIVITY (i)[Equation goes here - download the original to see it.] for all[Equation goes here - download the original to see it.] (ii)[Equation goes here - download the original to see it.] for all[Equation goes here - download the original to see it.]
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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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