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Linear Algebra, Complex Vector Spaces

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This chapter requires that we factor polynomials.
Of course, many polynomials do not factor over the real numbers;
for instance,
{x}^{2} + 1 does not factor into the product of two linear polynomials
with real coefficients.
For that reason, we shall from now on take our scalars from the complex
numbers.

That is, we are shifting from studying vector spaces over the real numbers
to vector spaces over the complex numbers—
in this chapter
vector and matrix entries are complex.

Any real number is a complex number and
a glance through this chapter shows that most of the examples use
only real numbers.
Nonetheless, the critical theorems require that the scalars be complex
numbers, so
the first section below is a quick review of complex numbers.

In this book we are moving to the more general context
of taking scalars to be complex only for the
pragmatic reason that we must do so in order to
develop the representation.
We will not go into using other sets of scalars in more detail because
it could distract from our goal.
However, the idea of taking scalars from a structure other than the real
numbers is an interesting one.
Delightful
presentations taking this approach are in
(Halmos 1958) and (Hoffman & Kunze 1971) harv error: no target: CITEREFHoffmanKunze1971 (help).

References

  • Halmos, Paul P. (1958), Finite Dimensional Vector Spaces (Second ed.), Van Nostrand.
  • Hoffman, Kenneth; Kunzy, Ray (1971), Linear Algebra (Second ed.), Prentice-Hall.

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