Linear Algebra Done Right · Problem 1B.8

Finite-Dimensional Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · complexification

Problem

Suppose

V

is a real vector space. • The complexification of

V

, denoted by

V_{\mathbf{C}}

, equals

V \times V

. An element of

V_{\mathbf{C}}

is an ordered pair

(u, v)

, where

u, v \in V

, but we write this as

u + iv

. • Addition on

V_{\mathbf{C}}

is defined by

(u_1 + iv_1) + (u_2 + iv_2) = (u_1 + u_2) + i(v_1 + v_2)

for all

u_1, v_1, u_2, v_2 \in V

. • Complex scalar multiplication on

V_{\mathbf{C}}

is defined by

(a + bi)(u + iv) = (au - bv) + i(av + bu)

for all

a, b \in \mathbf{R}

and all

u, v \in V

. Prove that with the definitions of addition and scalar multiplication as above,

V_{\mathbf{C}}

is a complex vector space. Think of

V

as a subset of

V_{\mathbf{C}}

by identifying

u \in V

with

u + i0

. The construction of

V_{\mathbf{C}}

from

V

can then be thought of as generalizing the construction of

\mathbf{C}^n

from

\mathbf{R}^n