Scalars, Linearity and their consequences
This portion of my web-space is in a state of flux (since 1997/Spring) while
I dismantle a huge page, add in some related fragments from other pages and set
the pieces together in such good order as I can manage, while at the same time
adjusting to many notational choices I've made since I wrote on this topic
before.
Thus far, I here describe:
addition 's place in the
sun, some context
and scalars .
rationals
and the continuum
The one-dimensional special case
which has its simplifications, yet shouldn't be over-simplified.
positivity the consequences of not having
an additive identity,
tensors , transposition
and trace three fundamental operations in
linear algebra; the page on the second now covers much of what the other two
cover.
the complex numbers and their
representation by a sphere .
hermitian conjugation combined with
transposition, consequent symmetries and
the two-dimensional special case .
Quadratic forms which provide,
among other things , a way to encode a notion
of length via a metric.
rotations a particular class of
isometry, preserving the lengths of a given metric.
unitary transformations associated with
hermitian forms; the example of SU(2)
projection the properties of
self-square linear maps – i.e., linear (V|f:V) for which
f·f=f,
projectors which are projections whose
composite with some metric-tensor is symmetric,
alternating algebra which weaves
antisymmetry into the world of smooth manifolds and
provides the basis for Hodge duality .
Lagrange's multipliers A method for
optimising functions of many variables, subject to constraints.
Euclidean spaces the classical flat
world and its relationship to linearity
Differentiation introduced as naturally
arising in the general linear context (in preference to arising for scalars
and being induced thereby for linear contexts).
The Fourier Transform is a way to
decompose a function as a linear combination of sinusoids.
ramblings sketches towards a better
approach, originally from my newer
area
a quick introduction to linearity
with a particular focus on addition (with further links)
and an old page on linearity , which I'm
shredding along with one on scalars . Other old
cruft include: mappings , binary operators
Eddy . 
Written by Eddy.