Page 108 - 35Linear Algebra

108 Vector Spaces

Complex numbers are very useful because of a special property that they
enjoy: every polynomial over the complex numbers factors into a product of
linear polynomials. For example, the polynomial

2
x + 1

doesn’t factor over real numbers, but over complex numbers it factors into

(x + i)(x − i) .

In other words, there are two solutions to

2
x = −1,

x = i and x = −i. This property has far-reaching consequences: often in
mathematics problems that are very difficult using only real numbers become
relatively simple when working over the complex numbers. This phenomenon
occurs when diagonalizing matrices, see chapter 13.
The rational numbers Q are also a field. This field is important in com-
puter algebra: a real number given by an infinite string of numbers after the
decimal point can’t be stored by a computer. So instead rational approxi-
mations are used. Since the rationals are a field, the mathematics of vector
spaces still apply to this special case.
Another very useful field is bits

B 2 = Z 2 = {0, 1} ,

with the addition and multiplication rules

+ 0 1 × 0 1
0 0 1 0 0 0
1 1 0 1 0 1

These rules can be summarized by the relation 2 = 0. For bits, it follows
that −1 = 1!
The theory of fields is typically covered in a class on abstract algebra or
Galois theory.

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