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Course by: Steven J. Cox. E-mail the author

# 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 n n and n n of columns of n n real and complex numbers, respectively.

To be more precise, we write xx and yy in n 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, zC z with xx as:
zx=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. 1x=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)=cx+cy 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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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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##### What are tags?

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks