Topology/Vector Spaces

Topology Vector Spaces Perfect Map Morphisms 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. Definition 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 100x100px|framed|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) Topology Vector Spaces Perfect Map Morphisms