Topology/Vector Spaces

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A vector space is formed by scalar multiples of vectors. The scalars are most commonly real numbers but can also be complex, or from any field.



A vector space   on a field   is a set   equipped with two binary operations: common vector addition on elements of   and scalar multiplication, by elements of   on elements of  . These operations are subject to 8 axioms (u,v, and w are vectors in   and a and b are scalars in  ):

1. Associativity (addition): (u+v)+w = u+(v+w).

2. Associativity (scalar and field multiplication): a(bu) = (ab)u

3. Distributivity (field addition): (a+b)u = au+bu

4. Distributivity (vector addition): a(u+v) = au+av

5. Identity element (addition):   0   such that u+0 = u   u  

6. Identity element (scalar multiplication): 1u = u

7. Commutativity: u+v = v+u

8. Inverse element:   -u   such that u+(-u) = 0   u  

An example of basic arrow vectors: first the black vector as the sum of the red and blue vectors, then the black vector as a sum of the blue vector and a scaling of the red vector (x2)

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