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Vector Space

Module by: Phil Schniter. E-mail the author

Summary: This module introduces vector space.

  • A vector space consists of the following four elements:
    1. A set of vectors VV,
    2. A field of scalars (where, for our purposes, is either R or C),
    3. The operations of vector addition "+" (i.e., + : V×VV V V V )
    4. The operation of scalar multiplication "⋅"(i.e., ⋅ : ×VV V V )
    for which the following properties hold. (Assume xyzV x y z V and αβ α β .)
Table 1
Properties Examples
commutativity x+y=y+x x y y x
associativity (x+y)+z=x+(y+z) x y z x y z
(αβ)x=α(βx) α β x α β x
distributivity α(x+y)=(αx)+(αy) α x y α x α y
(α+β)x=αx+βx α β x α x β x
additive identity 0 ,0V:x+0=x  ,   xV    x V 0 0 V x 0 x
additive inverse x ,(x)V:x+x=0  ,   xV    x V x x V x x 0
multiplicative identity 1x=x  ,   xV    x V 1 x x

Important examples of vector spaces include

Table 2
Properties Examples
real NN-vectors V=RN V N , =R
complex NN-vectors V=CN V N , =C
sequences in " lp lp " V= xn nZ:n=|xn|p< V x n n n x n p , =C
functions in " p p " V= ft |ft|pdt< V f t t f t p , =C
where we have assumed the usual definitions of addition and multiplication. From now on, we will denote the arbitrary vector space (VV, , +, ⋅) by the shorthand VV and assume the usual selection of (, +, ⋅). We will also suppress the "⋅" in scalar multiplication, so that αx α x becomes αx α x .

  • A subspace of VV is a subset MV M V for which
    1. (x+y)M  ,   xMyM    x y x M y M x y M
    2. αxM  ,   xMα    x M α α x M

    note:

    Note that every subspace must contain 00, and that VV is a subspace of itself.
  • The span of set SV S V is the subspace of VV containing all linear combinations of vectors in SS. When S=x0x N - 1 S x0 x N - 1 , spanS i =0N1 α i x i α i span S i 0 N 1 α i x i α i
  • A subset of linearly-independent vectors x0x N - 1 V x0 x N - 1 V is called a basis for V when its span equals VV. In such a case, we say that VV has dimension NN. We say that VV is infinite-dimensional 1 if it contains an infinite number of linearly independent vectors.
  • VV is a direct sum of two subspaces MM and NN, written V=MN V M N , iff every xV x V has a unique representation x=m+n x m n for mM m M and nN n N .

    note:

    Note that this requires MN=0 M N 0

Footnotes

  1. The definition of an infinite-dimensional basis would be complicated by issues relating to the convergence of infinite series. Hence we postpone discussion of infinite-dimensional bases until the Hilbert Space section.

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