Vector Spaces
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Properties of vector spaces
For any vector space, , and for every vector and number then (i)[Equation goes here - download the original to see it.] (ii)[Equation goes here - download the original to see it. (iii)[Equation goes here - download the original to see it.] (iv)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.] (v) 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.] Proof (i) Since[Equation goes here - download the original to see it.]by axiom B4, we have (with[Equation goes here - download the original to see it.]) [Equation goes here - download the original to see it. Adding[Equation goes here - download the original to see it.]to both sides (which exists, by axiom A3), we have [Equation goes here - download the original to see it.] But [Equation goes here - download the original to see it.], by axiom A3. So [Equation goes here - download the original to see it.] (ii) Since[Equation goes here - download the original to see it.]by axiom B4, we have (with[Equation goes here - download the original to see it.]) [Equation goes here - download the original to see it.] so that[Equation goes here - download the original to see it.], by axiom A2. Adding[Equation goes here - download the original to see it.]to both sides (which exists, by axiom A3), we have [Equation goes here - download the original to see it.] But [Equation goes here - download the original to see it.], by axiom A3. So [Equation goes here - download the original to see it.] (iii) To show that [Equation goes here - download the original to see it.], we first prove that[Equation goes here - download the original to see it.] Now [Equation goes here - download the original to see it.] It follows, on adding[Equation goes here - download the original to see it.], to both sides, that [Equation goes here - download the original to see it.] by axiom A2 So [Equation goes here - download the original to see it.]by axiom A4 that is [Equation goes here - download the original to see it.]by axiom A3 So [Equation goes here - download the original to see it.] by axiom A2
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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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