Vector Spaces
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Subspaces of vector spaces
A vector subspace is a vector space within a vector space. A formal definition is If is a vector space and is a non-empty subset of then is a subspace of if (1) 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.] (2) 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.]. It can be shown that the other vector axioms hold of a vector subspace provided these two conditions hold, so a vector subspace is itself a vector space. Example: The set[Equation goes here - download the original to see it.]subject to the condition[Equation goes here - download the original to see it.]is a subspace of the vector space of linear polynomials. To show this we prove that the two conditions in the definition of a vector subspace are satisfied. (1)Let [Equation goes here - download the original to see it.], then[Equation goes here - download the original to see it.]where[Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.] where[Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it.]and [Equation goes here - download the original to see it.] so [Equation goes here - download the original to see it.] so 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.]satisfies[Equation goes here - download the original to see it.] and [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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