Vector Spaces

Overview


Vector spaces are ubiquitous in mathematics and physics. They are defined to be a collection of object equipped with an ability to add two objects, and to multiply any object by a number. Examples include the real numbers, complex numbers, column vectors.

Axioms


  1. Closure under Addition
    For every x and y in V, there is a unique element in V, x+y
  2. Associativity of Addition
    (x+y)+z = x+(y+z)
  3. Closure under Scalar Mulitplication
    For every elment x in V, and number a in the field F, there is a unique element in V, ax.
  4. Existence of Zero Vector
    There exists a zero vector, 0, such that x+0=0+x=x
  5. Existence of Inverse
    For every vector x, there is another vector,-x, which added to x yields the zero vector.

Topics


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