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Vector Space

Module by: Phil Schniter

Summary: This module introduces vector space.

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  • 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 or ),
    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
αβx=αβx α β x α β x
distributivity αx+y=αx+αy α x y α x α y
α+βx=αx+βx α β x α x β x
additive identity ,xV:0,0V:x+0=x x V 0 0 V x 0 x
additive inverse ,xV:-x,-xV:x+-x=0 x V x x V x x 0
multiplicative identity ,xV:1x=x x V 1 x x

Important examples of vector spaces include

Table 2
Properties Examples
real NN-vectors V=N V N , =
complex NN-vectors V=N V N , =
sequences in " lp lp " V={xn|,n:n=-|xn|p<} V x n n n x n p , =
functions in " p p " V={ft|-|ft|pdt<} V f t t f t p , =
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,xMyM:x+yM x y x M y M x y M
    2. ,xMα:αxM 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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