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【转载】Vector Spaces  

2017-10-10 08:34:20|  分类: 线性代数 |  标签: |举报 |字号 订阅

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本文转载自jia_huiqiang《Vector Spaces》

Definition

A vector space V is a set with two operations + and * that satisfy the following properties.

 a.  If u and v are elements of V, then u + v is and element of V (closure under +)

  1. u + v  =  v + u
  2. u + (v + w)  =  (u + v) + w
  3. There is an element 0 in V such that
              u + 0  =  0 + u  =  u 
  4. For every u in V there is an element -u with
              u + (-u)  =  0 

b.  If u is in V and c is a real number then c*u is in V (closure under *)

  1. c * (u + v)  =  c * u + c * v 
  2. (c + d) * u  =  c * u + d * u 
  3. c * (d * u)  =  (cd) * u 
  4. 1 * u  =  u

 

You will recognize these properties as properties of vectors in Rn, however there is a large class of vector spaces that do not look at all like Rn.  

Examples of Vector Spaces

Example 1  P2 

Consider the set P2 of polynomials of degree less than or equal to 2.  Define + to be polynomial addition

        (a1t2 + b1t + c2)  +  (a2t2 + b2t + c2)  =  (a1 + a2)t2 + (b1 + b2)t + (c1 + c2)

and * is defined by

        k * (at2 + bt + c)  =  (ak)t2 + (kb)t + (kc)

This is a vector space.  Most of the properties clearly hold.  We will demonstrate a few of the properties.  For example the 0 vector is the zero polynomial (0).  We have

        (at2 + bt + c) + 0  =  0 + (at2 + bt + c)  =  at2 + bt + c

Property b2 holds since 

        (r + s) * (at2 + bt + c)  =  (r + s)at2 + (r + s)bt + (r + s) c

        =  (ra + sa)t2 + (rb + sb)t + (rc + sc)  =  (ra)t2 + (sa)t2 + (rb)t + (sb)t + rc + sc

        =  (ra)t2  + (rb)t + rc + (sa)t2+ (sb)t + sc  =  r(at2 + bt + c) + s(at2 + bt + c)

Example 2  Pn 

We can generalize Example 1 and let Pn be the set of all polynomials of degree less than n.  We define + to mean polynomial addition and * to be scalar multiplication as in Example 1.  This is a vector space as you can check.

Example 3  M2x3 

Consider the set M2x3 of 2 x 3 matrices and let + be defined by matrix addition and * be defined by matrix scalar multiplication.  Then M2x3 is a vector space.  We have stated all of the required properties previously.

Example 4  Mmxn 

We can generalize Example 3 by letting Mmxn be the set of all m x n matrices with matrix addition and scalar multiplication as before.  

Example 5

Consider the set V of all differentiable functions f such that f '(1)  =  0.  Let + be defined as addition of functions and * be defined as regular scalar multiplication.  This is a vector space.  We will demonstrate a few of the properties.  Let f and g be elements of this set.  Then 

        f '(1)  =  g '(1)  =  0

To show additive closure, we have

        (f + g)'(1)  =  f '(1) + g'(1)  =  0 + 0  =  0

so that f + g is in V.  To show multiplicative closure we have

        (cf)'(1)  =  c(f '(1))  =  c(0)  =  0

The rest of the properties follow from the properties of function arithmetic and derivatives.  

Example 6

Let S be the set of ordered pairs in R2 with + defined by

        (x1, y1) + (x2, y2)  =  (x1 + 2x2, y1 + 2y2)

and * defined by

        c(x,y)  =  (cx,cy)

then S is not a vector space, since property a1 fails.  For example

        (2,3) + (4,5)  =  (10,13)

but 

        (4,5) + (2,3)  =  (8,11)

A Few Properties of Vector Spaces

Vector spaces enjoy several additional properties that we will later explore.  Below are some of the most basic ones.

Theorem

Let V be a vector space then

  1. 0u  =  0  for all u in V
  2. c0  =  0  for all scalars c
  3. If cu  =  0  then either c  =  0 or u  =  0
  4. (-1)u  =  -u  for all u in V

 

We will prove 4.  We have

        (-1)u + u  =  (-1)u + (1)u  

        =  (-1 + 1)u  =  0u  =  0

Since

        (-1)u + u  =  0 

We can conclude that (-1)u is the additive inverse of u.  

reference:http://ltcconline.net/greenl/courses/203/Vectors/vectorSpaces.htm

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