Page 83 - Linear Algebra Done Right

Real Coefficients
Real Coefficients
Before discussing polynomials with real coefficients, we need to 69
learn a bit more about the complex numbers.
Suppose z = a + bi, where a and b are real numbers. Then a is
called the real part of z, denoted Re z, and b is called the imaginary
part of z, denoted Im z. Thus for every complex number z, we have
z = Re z + (Im z)i.

The complex conjugate of z ∈ C, denoted ¯ z, is defined by Note that z = ¯ z if and
only if z is a real
¯ z = Re z − (Im z)i.
number.
For example, 2 + 3i = 2 − 3i.
The absolute value of a complex number z, denoted |z|, is defined
by

2
2
|z|= (Re z) + (Im z) .
√
For example, |1 + 2i|= 5. Note that |z| is always a nonnegative
number.
You should verify that the real and imaginary parts, absolute value,
and complex conjugate have the following properties:
additivity of real part
Re(w + z) = Re w + Re z for all w, z ∈ C;
additivity of imaginary part
Im(w + z) = Im w + Im z for all w, z ∈ C;

sum of z and ¯ z
z + ¯ z = 2Re z for all z ∈ C;

difference of z and ¯ z
z − ¯ z = 2(Im z)i for all z ∈ C;

product of z and ¯ z
2
z¯ z =|z| for all z ∈ C;
additivity of complex conjugate
w + z = ¯ w + ¯ z for all w, z ∈ C;

multiplicativity of complex conjugate
wz = ¯ w ¯ z for all w, z ∈ C;