Lecture Notes in Differential Equations - Bruce E. Shapiro

128 LESSON 15. LINEAR OPERATORS AND VECTOR SPACES
5. Additive Inverse. For each y ∈ V there exists a −y ∈ V such that
y + (−y) = (−y) + y = 0 (15.5)
6. Multiplicative Identity. For every y ∈ V,
1 × y = y × 1 = y (15.6)
7. Distributive Property. For every a, b ∈ R and every y, z ∈ V,
a(y + z) = ay + az
(a + b)y = ay + by
(15.7)
Example 15.1. The usual Euclidean 3D space forms a vector space, where
each vector is a triple of numbers corresponding to the coordinates of a point
If w = (p, q, r) then addition of vectors is defined as
and scalar multiplication is given by
v = (x, y, z) (15.8)
v + w = (x + p, y + q, r + z) (15.9)
You should verify that all seven properties hold.
av = (ax, ay, az) (15.10)
We are particularly interested in the following vector space.
Example 15.2. Let V be the set of all functions y(t) defined on the real
numbers. Then V is a vector space under the usual definitions of addition
of functions and multiplication by real numbers. For example, if f and g
are functions in V then
h(t) = f(t) + g(t) (15.11)
is also in V, and if c is a real number, then
p(t) = cf(t) (15.12)
is also in V. To see that the distributive property holds, observe that
a(f(t) + g(t)) = af(t) + bg(t) (15.13)
(a + b)f(t) = af(t) + bft(t) (15.14)
You should verify that each of the other six properties hold.