Note on Conventions
Throughout this book we follow the convention (customary among topologists)
of writing the operator before the operand, and thus writing compositions from
right to left. Since the linear (and quadratic) algebra in this book is intimately
related to the topology, we are forced to adopt the corresponding conventions
there, with the following consequences.
Scalar multiplication in a module (which commutes with linear operators) is
written on the right, thus we habitually study right modules. Given a linear
map between free modules
a: V ^ W
where {e^} is a basis of V and {fj} a basis of W, we write
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and make a correspond to the matrix A with (A)ji = a,ji. Frequently we denote
an operator and its matrix by the same symbol, when the bases are understood.
We also use matrix notation more generally for maps into or from a direct sum
of modules, as e.g. in
h1}
: V-*W1®W2 , (h b2) : W1®W2-X.
The composite map is to be evaluated by the usual rule for matrix products,
not forgetting that we write composites from right to left. Thus, for example,
the composite of the two maps above is
(61 M ( ^ ) = M 1 + M 2 : V^X.
For our sign conventions see the beginning of §2.
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