A more heterogeneous example of a vector space is data tracks.
A depth-sounding survey of a lake can make a vector space that is
a collection of tracks,
a vector of vectors
(each vector having a different number of components,
because lakes are not square).
This vector space of depths along tracks in a lake
contains the depth values only.
The (*x*,*y*)-coordinate information
locating each measured depth value
is (normally) something outside the vector space.
A data space could also be a collection of echo soundings,
waveforms recorded along tracks.

We briefly recall information about vector spaces found in elementary books:
Let be any scalar.
Then if is a vector and is conformable
with it, then other vectors are
and .The size measure of a vector is a positive value called a norm.
The norm is usually defined to be the dot product
(also called the *L _{2}* norm), say .For complex data it is
where is the complex conjugate of .In theoretical work the ``length of a vector'' means the vector's norm.
In computational work the ``length of a vector'' means the
number of components in the vector.

Norms generally include a weighting function.
In physics,
the norm generally measures a conserved quantity
like energy or momentum,
so, for example,
a weighting function for magnetic flux is permittivity.
In data analysis,
the proper choice of the weighting function is
a practical statistical issue,
discussed repeatedly throughout this book.
The algebraic view of a weighting function is that
it is a diagonal matrix
with positive values spread along the diagonal,
and it is denoted .With this weighting function
the *L _{2}* norm of a data space is denoted
.Standard notation for norms uses a double absolute value,
where .A central concept with norms is the triangle inequality,
whose proof you might recall (or reproduce with the use of dot products).

4/27/2004