Page 69 - Advanced Linear Algebra

Vector Spaces 53

rrk²(³ ~ dim ² rs²(³³ ~ dim ² rs²(,³³ ~ rrk²(,³
as desired.

, then rrk ²(³ ~ crk . This number is called the
Theorem 1.16 If ( C Á ²(³
(
rank of and is denoted by ² rk ( . ³
Proof. According to the previous lemma, we may reduce to reduced column
(
echelon form without affecting the row rank. But this reduction does not affect
the column rank either. Then we may further reduce to reduced row echelon
(
form without affecting either rank. The resulting matrix 4 has the same row
and column ranks as . But 4 is a matrix with 's followed by 's on the main
(

(
diagonal entries 4Á 4Á Ã ) and 's elsewhere. Hence,

Á
Á
rrk²(³ ~ rrk²4³ ~ crk²4³ ~ crk²(³
as desired.
The Complexification of a Real Vector Space
(
If > is a complex vector space that is, a vector space over ) , then we can
d
think of > as a real vector space simply by restricting all scalars to the field .
s
Let us denote this real vector space by > and call it the real version of > s .
On the other hand, to each real vector space , we can associate a complex
=
vector space = d . This “complexification” process will play a useful role when
we discuss the structure of linear operators on a real vector space. Throughout
(
our discussion will denote a real vector space.)
=
=
Definition If is a real vector space, then the set = d ~ = d = of ordered
pairs, with componentwise addition
²"Á#³ b ²%Á&³ ~ ²" b %Á# b &³
and scalar multiplication over defined by
d
² b ³²"Á #³ ~ ² " c #Á # b "³
=
for Á  s is a complex vector space, called the complexification of .
It is convenient to introduce a notation for vectors in = d that resembles the
notation for complex numbers. In particular, we denote ²"Á #³  = d by " b #
and so

= d ~ ¸ " b # “ " Á #  = ¹
Addition now looks like ordinary addition of complex numbers,
²" b# ³ b²%b& ³ ~ ²" b%³b²#b&³

and scalar multiplication looks like ordinary multiplication of complex numbers,