Theory of Statistics - George Mason University

630 0 Statistical Mathematics
0.0.4.2 Subsets of Linear Spaces
Interesting subsets of a given linear space Ω are formed as {x ; x ∈ Ω, g(x) =
0}, for example, the plane in IR 3 defined by c1x1 + c2x2 + c3x3 = c0 for some
constants c0, c1, c2, c3 ∈ IR and where (x1, x2, x3) ∈ IR 3 .
Many subsets of IR d are of interest because of their correspondence to
familiar geometric objects, such as lines, planes, and structures of finite extent
such as cubes, and spheres. An object in higher dimensions that is analogous
to a common object in IR 3 is often called by the name of the three-dimensional
object preceded by “hyper-”; for example, hypersphere.
A subset of a linear space may or may not be a linear space. For example,
a hyperplane is a linear space only if it goes through the origin. Other subsets
of a linear space may have very little in common with a linear space, for
example, a hypersphere.
A hyperplane in IR d is often of interest in statistics because of its use
as a model for how a random variable is affected by covariates. Such a linear
manifold in IR d is determined by d points that have the property of affine independence.
A set of elements x1, . . ., xd ∈ Ω are said to be affinely independent
if x2 − x1, . . ., xd − x1 are linearly independent. Affine independence in this
case insures that the points do not lie in a set of less than d −1 dimensions.
0.0.4.3 Inner Products
We now define a useful real-valued binary function on linear spaces.
Definition 0.0.7 (inner product)
If Ω is a linear space, an inner product on Ω is a real-valued function, denoted
by 〈x, y〉 for all x and y in Ω, that satisfies the following three conditions for
all x, y, and z in Ω.
1. Nonnegativity and mapping of the identity:
if x = 0, then 〈x, x〉 > 0 and 〈0, x〉 = 〈x, 0〉 = 〈0, 0〉 = 0.
2. Commutativity:
〈x, y〉 = 〈y, x〉.
3. Factoring of scalar multiplication in inner products:
〈ax, y〉 = a〈x, y〉 for real a.
4. Relation of vector addition to addition of inner products:
〈x + y, z〉 = 〈x, z〉 + 〈y, z〉.
Inner products are often called dot products, although “dot product” is
often used to mean a specific inner product.
A linear space together with an inner product, (Ω, 〈·, ·〉), is called an inner
product space.
A useful property of inner products is the Cauchy-Schwarz inequality:
Theory of Statistics c○2000–2013 James E. Gentle