Page 17 - Linear Algebra Done Right

Complex Numbers
distributive property
λ(w + z) = λw + λz for all λ, w, z ∈ C.
For z ∈ C, we let −z denote the additive inverse of z. Thus −z is 3
the unique complex number such that
z + (−z) = 0.
Subtraction on C is defined by

w − z = w + (−z)

for w, z ∈ C.
For z ∈ C with z = 0, we let 1/z denote the multiplicative inverse
of z. Thus 1/z is the unique complex number such that

z(1/z) = 1.

Division on C is defined by

w/z = w(1/z)
for w, z ∈ C with z = 0.
So that we can conveniently make definitions and prove theorems
that apply to both real and complex numbers, we adopt the following
notation:
Throughout this book, The letter F is used
F stands for either R or C. because R and C are
examples of what are
Thus if we prove a theorem involving F, we will know that it holds when called fields. In this
F is replaced with R and when F is replaced with C. Elements of F are book we will not need
called scalars. The word “scalar”, which means number, is often used to deal with fields other
when we want to emphasize that an object is a number, as opposed to than R or C. Many of
a vector (vectors will be defined soon). the definitions,
For z ∈ F and m a positive integer, we define z m to denote the theorems, and proofs
product of z with itself m times: in linear algebra that
work for both R and C
z m = z · ··· · z .
also work without
m times change if an arbitrary
m n
m m
Clearly (z ) = z mn and (wz) m = w z for all w, z ∈ F and all field replaces R or C.
positive integers m, n.