Definition: Linear Space
Let's begin by abstracting what we have done in the two cases above.
The first is the easiest to think about. In the case of describing an object
moving in three space, we have started with three objects for which we know some
physical properties or have some intuitions about: the three basis directions.
We then multiplied each direction be a length and added the resulting three vectors
formally. We can abstract what we got as follows. A linear space (or a vector
space) is a set of elements (vectors), V, and a set of numbers (scalars), S,
(where for us, S will be either the real or complex numbers) satisfying the following
properties: (Click here for a definition of the specialized math symbols used.)
- The set forms a group under addition. This
means: it is closed, an identity exists, and each element has an inverse.
Written out mathematically:
- If a, b ∈
V, then a + b ∈ V. (Note
this means we have some definition of +).
- There exists a vector 0, such that
for any a ∈ V, a + 0 = 0 + a = a.
every a ∈ V, ∃ (-a) ∈ V such
that a + (-a) = (-a) + a =
set forms a group under multiplication by a scalar. This
means: it is closed, there is an identity,
and every scalar has an inverse (except 0). Written
- If a ∈ V and α ∈ S,
then αa ∈ V. (Note this
means we have a definition of what it means to multiply a vector
by a scalar).
- There exists a scalar 1
such that for any a ∈ V, 1⋅a=a⋅1
- For every &alpha: ∈ S (except for α = 0), ∃ α-1 ∈ S, such that α-1(αa) = α(α-1a) = a.
- The two operations of addition and scalar multiplication are distributive. Mathematically, this means that addition and multiplication work like for regular numbers.
- ∀ α ∈ S, a, b ∈ V, α(a + b) = (αa) +(αb) and
∀ α, β ∈ S, a ∈ V, (α + β)a = (αa) +(βa)
These properties specify a linear (or vector) space.
It is easy to show that the two examples discussed in
the Motivation section both satisfy all these properties.
|University of Maryland||Physics Department||Physics 374 Home
This page prepared by
Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120
Last revision 25. October, 2005.