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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 ℝ or ℂ),
    3. The operations of vector addition "+" (i.e., + : V×V→V V V V )
    4. The operation of scalar multiplication "⋅"(i.e., ⋅ : ×V→V V V )
    for which the following properties hold. (Assume x∧y∧z∈V x y z V and α∧β∈ 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 α⋅x+y=α⋅x+α⋅y α ⋅ x y ⋅ α x ⋅ α y
α+βx=αx+βx α β x α x β x
additive identity ∀,x∈V:∃0,0∈V:x+0=x x V 0 0 V x 0 x
additive inverse ∀,x∈V:∃-x,-x∈V:x+-x=0 x V x x V x x 0
multiplicative identity ∀,x∈V:1⋅x=x 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|∃,n∈ℤ:∑n=-∞∞|xn|p<∞} V x n n n x n p , =ℂ
functions in " ℒp ℒp " 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 α⋅x ⋅ α x becomes αx α x .

  • A subspace of VV is a subset M⊂V M V for which
    1. ∀x,y,x∈M∧y∈M:x+y∈M x y x M y M x y M
    2. ∀,x∈M∧α∈:αx∈M 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 S⊂V S V is the subspace of VV containing all linear combinations of vectors in SS. When S=x0…x N - 1 S x0 … x N - 1 , spanS≔{∑i=0N−1 α i x i | α i ∈} ≔ span S i 0 N 1 α i x i α i
  • A subset of linearly-independent vectors x0…x 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=M⊕N V M N , iff every x∈V x V has a unique representation x=m+n x m n for m∈M m M and n∈N n N .

    note:

    Note that this requires M⋂N=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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