Basic operators and adjoints |

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 -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 **norm**), say
.
For complex data it is
,
where
is the complex conjugate of .
A notation that does transpose and complex conjugate at the same time is
.
In theoretical work, the ``size of a vector'' means the vector's norm.
In computational work the ``size 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;
therefore, 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 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,
a proof you might recall (or reproduce with the use of dot products).

Basic operators and adjoints |

2014-09-27