Kunze Linear Algebra · Problem 3.3.5

Linear Transformations

Mathematics — Methods · complex-numbers-matrix-representation

Problem

Let

V

be the set of complex numbers regarded as a vector space over the field of real numbers (Exercise 1). We define a function

T

from

V

into the space of

2 \times 2

real matrices, as follows. If

z = x + iy

with

x

and

y

real numbers, then

T(z) = \begin{bmatrix} x + 7y & 5y \\ -10y & x - 7y \end{bmatrix}.

(a) Verify that

T

is a one-one (real) linear transformation of

V

into the space of

2 \times 2

real matrices. (b) Verify that

T(z_1 z_2) = T(z_1)T(z_2)

. (c) How would you describe the range of

T