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Vector Spaces
61. Notation: Let IR denote the set of real numbers and let Cl denote the set of complex
numbers. A scalar is a real or complex number.
62. Definition (Real Vector Space) A real vector space V is a nonempty set of objects, called
vectors, together with a definition of addition and scalar multiplication, that satisfy the
following axioms. If u, v, w ∈ V and α, β ∈ IR, then
(1) u + v ∈ V (closure under addition),
(2) αv ∈ V (closure under scalar multiplication),
(3) u + v = v + u,
(4) (u + v) + w = u + (v + w),
(5) there is a zero vector, 0, so that u + 0 = u,
(6) there is a vector −u ∈ V so that u + (−u) = 0,
(7) α(βu) = (αβ)u,
(8) (α + β)u = αu + βu,
(9) α(u + v) = αu + αv,
(10) 1u = u.
63. Remark: The definition of a complex vector space is the same except that the scalars are
complex.
64. Theorem (Vector Space) Let V be a vector space, let x ∈ V and let α be a scalar. Then
the following hold.
• α0 = 0.
• 0x = 0.
• If αx = 0, then α = 0, or x = 0, or both.
• (−1)x = −x.
65. Examples: IRm×n , Cl m×n , 5n , 5, and C[a, b].
66. Definition (Inner or Dot Product) Let V be a complex vector space. An inner (dot)
product on V is a function satisfying the following axioms. Let u, v, w ∈ V and let α
be a scalar. Then
• u · v is a scalar,
• (u + v) · w = u · w + v · w,
• u · v = v · u,
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• (αu) · v = α(u · v),
• u · u ≥ 0 and u · u = 0 if and only if u = 0.
67. Examples: Usual inner products.
• IRn×1 .
• MTHSC 206 n-space.
• IRn .
68. Definition (Norm) Let V be a vector space. A norm on V is a function that satisfies the
following axioms. Let u, v ∈ V and let α be a scalar. Then
• kuk ≥ 0 and kuk = 0 implies u = 0,
• ku + vk ≤ kuk + kvk,
• kαuk = |α|kuk.
69. Examples: Vector norms.
• Norm induced by an inner product.
• kxk p . MATLAB: norm(v, p).
70. Examples: Matrix norms.
• k Ak p . MATLAB: norm(A,p).
• k Ak F . MATLAB: norm(A,’fro’).
71. Definition (Subspace) Let F be a subset of a vector space V . If F is a vector space using
the same operations defined for V , then F is called a subspace of V .
72. Theorem (Test for Subspace). A subset F of a vector space V is a subspace if and only
if F is closed under addition and scalar multiplication (satisfies vector space axioms 1
and 2).
73. Remark: Given a subset F of a vector space V , is F a subspace? Answer:
Yes if vector space axioms 1 and 2 are satisfied.
No if any vector space axiom is not satisfied.
74. Definitions (Linear Combination and Span) Let u1 , . . . , uk ∈ V , a vector space, and let
α1 , . . . , αn be scalars. The expression
α1 u1 + · · · + αk uk
is called a linear combination of u1 , . . . , uk . The span of u1 , . . . , uk is defined to be the
set of all linear combinations of the vectors u1 , . . . , uk . We write span{u1 , . . . , uk }.
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75. Theorem (Span) If u1 , . . . , uk ∈ V , a vector space, then span{u1 , . . . , uk } is a subspace
of V .
76. Definition (Null Space) Suppose A ∈ IRm×n . The null space of A is defined by
N( A) = {x ∈ IRn×1 : Ax = 0}.
77. Definition (Range) Suppose A ∈ IRm×n . The range of A is defined by
Ran( A) = {y ∈ IRm×1 : y = Ax for some x ∈ IRn×1 }.
78. Theorem N( A) is a subspace of IRn×1 and Ran( A) is a subspace of IRm×1 .
79. Remark: N( A) is the solution set for the homogeneous system Ax = 0. We already
know how to compute it.
80. Let A = [a1 , . . . , an ]. Then
y = Ax =
a1

x1


. . . an  ...  = x1 a1 + · · · + xn an .
xn

so y ∈ Ran( A) if and only if y is a linear combination of the columns of A. Ran( A) is
sometimes called the column space of A.
81. Definition (Linear Dependence) The vectors v1 , . . . , vk in a vector space V are linearly
dependent if there are scalars α1 , . . . , αk , not all zero, so that
α1 v1 + . . . + αk vk = 0.
82. Definition (Linear Independence) The vectors v1 , . . . , vk in a vector space V are linearly
independent if the vector equation
α1 v1 + · · · + αk vk = 0
implies that α1 = α2 = · · · = αk = 0.
83. Problem: Determine if a set of vectors is linearly dependent or linearly independent.
84. Definition (Finitely Generated Vector Space) A vector space V is said to be finitely
generated if there exists a finite set of vectors v1 , . . . , vn so that
V = span{v1 , . . . , vn }.
85. Definition (Basis) A set of vectors v1 , . . . , vn is said to be a basis for a finitely generated
vector space V if
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• V = span{v1 , . . . , vn },
• v1 , . . . , vn are linearly independent.
86. A ref for a matrix A can be used to compute bases for N( A) and Ran( A). A null space
example.
87. Theorem All bases for nontrivial finitely generated vector space have the same number
of elements.
88. Definition (Dimension) If V is a trivial vector space, its dimension is defined to be 0; if
V is a finitely generated vector space, its dimension is defined to be the (unique) number
of vectors in a basis. Notation: dim(V ). If a vector space V is not finitely generated, it
is said to be infinite-dimensional.
89. Examples.
90. Remark: rank(A) is
• the number of pivots in a ref for A,
• the number of dependent variables in the system Ax = b,
• dim(Ran( A)),
• the maximum number of linearly independent columns in A.
91. nullity(A) is
• the column dimension of A - rank( A),
• the number of free variables in the system Ax = b,
• dim(N( A)).
92. Visualization of subspaces and dimension.
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