# 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:

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.
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).
ramblings
sketches towards a better approach, originally from my newer area
a quick introduction