Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 15
Linear Operators and
Vector Spaces
Definition 15.1. A Vector Space over R 1 is a set V combined with two
operations addition (denoted by x + y, y, y ∈ V) and scalar 2 multiplication
(denoted by c × y or cy, x ∈ R, y ∈ V ). with the following properties:
1. Closure under Addition and Scalar Multiplication
y, z ∈ V =⇒ y + z ∈ V
t ∈ R, y ∈ V =⇒ ty ∈ V
(15.1)
2. Commutativity of Addition
y, z ∈ V =⇒ y + z = z + y (15.2)
3. Associativity of Addition and Scalar Multiplication
w, y, z ∈ V =⇒ (w + y) + z = w + (y + z)
a, b ∈ R, y ∈ V =⇒ (ab)y = a(by)
(15.3)
4. Additive Identity. There exists a 0 ∈ V such that
y ∈ V =⇒ y + 0 = 0 + y = y (15.4)
1 This definition generalizes with R replaced by any field.
2 A scalar is any real number or any real variable.
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