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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×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
(α⁢β)⁢x=α⁢(β⁢x) α β x α β x
distributivity α⁢⋅⁢(x+y)=(α⋅x)+(α⋅y) α ⋅ x y ⋅ α x ⋅ α y
(α+β)⁢x=α⁢x+β⁢x 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= x⁢n ∃n∈Z:∑n=−∞∞|x⁢n|p<∞ V x n n n x n p , =C
functions in " ℒp ℒp " V= f⁢t ∫−∞∞|f⁢t|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 α⁢x 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∈:α⁢x∈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 , span⁢S≔ ∑ 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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