Page 70 - Advanced Linear Algebra

54 Advanced Linear Algebra

² b ³²" b# ³ ~ ² " c #³ b² # b "³
Thus, for example, we immediately have for Á  s ,
²" b # ³ ~ " b #
²" b # ³ ~ c # b "
² b ³" ~ " b "
² b ³# ~ c # b #

The real part of '~ " b # is "  = and the imaginary part of is #  = .
'
The essence of the fact that '~ " b #  = d is really an ordered pair is that is
'
if and only if its real and imaginary parts are both .
We can define the complexification map cpx¢= ¦ = d by
cpx²#³ ~ # b

Let us refer to #b as the complexification , or complex version of #  = .
Note that this map is a group homomorphism, that is,
cpx² ³ ~ b and cpx²" f #³ ~ cpx²"³ f cpx²#³

and it is injective:
cpx²"³ ~ cpx²#³ ¯ " ~ #
Also, it preserves multiplication by real scalars:

cpx² "³ ~ " b ~ ²" b ³ ~ cpx²"³
for  s . However, the complexification map is not surjective, since it gives
only “real” vectors in = d .
The complexification map is an injective linear transformation defined in the
(
)
next chapter from the real vector space to the real version ² = d ³ s of the
=
complexification = d , that is, to the complex vector space = d provided that
scalars are restricted to real numbers. In this way, we see that = d contains an
embedded copy of .
=
The Dimension of = d
The vector-space dimensions of = and = d are the same. This should not
necessarily come as a surprise because although = d may seem “bigger” than ,
=
the field of scalars is also “bigger.”
Theorem 1.17 If 8 is a basis for = over s ~¸# “ 0¹ , then the
complexification of ,
8
cpx²³ ~ ¸# b “ #  ¹
8
8