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Vector Space

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

Summary: This module discusses vector spaces and their applications to complex arithmetic.

Introduction

You have long taken for granted the fact that the set of real numbers, R, is closed under addition and multiplication, that each number has a unique additive inverse, and that the commutative, associative, and distributive laws were right as rain. The set, C, of complex numbers also enjoys each of these properties, as do the sets ℝ n ℝ n and ℂ n ℂ n of columns of n n real and complex numbers, respectively.

To be more precise, we write xx and yy in ℝ n ℝ n as

x= x 1 x 2 … x n T x x 1 x 2 … x n

y= y 1 y 2 … y n T y y 1 y 2 … y n

and define their vector sum as the elementwise sum

x+y= x 1 + y 1 x 2 + y 2 ⋮ x n + y n x y x 1 y 1 x 2 y 2 ⋮ x n y n
(1)
and similarly, the product of a complex scalar, z∈C z with xx as:
z⁢x=z⁢ x 1 z⁢ x 2 ⋮z⁢ x n z x z x 1 z x 2 ⋮ z x n
(2)

Vector Space

These notions lead naturally to the concept of vector space. A set V V is said to be a vector space if

  1. x+y=y+x x y y x for each x x and y y in V V
  2. x+y+z=x+y+z x y z x y z for each x x, y y, and z z in V V
  3. There is a unique "zero vector" such that x+0=x x 0 x for each x x in V V
  4. For each x x in V V there is a unique vector −x x such that x+−x=0 x x 0 .
  5. 1⁢x=x 1 x x
  6. ( c 1 ⁢ c 2 )⁢x= c 1 ⁢( c 2 ⁢x) c 1 c 2 x c 1 c 2 x for each x x in V V and c 1 c 1 and c 2 c 2 in C .
  7. c⁢(x+y)=c⁢x+c⁢y c x y c x c y for each x x and y y in V V and c c in C .
  8. ( c 1 + c 2 )⁢x= c 1 ⁢x+ c 2 ⁢x c 1 c 2 x c 1 x c 2 x for each x x in V V and c 1 c 1 and c 2 c 2 in C .

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