1. Motivated by Example 2, let us define a function or trans- formation T from the set of all complex numbers to the space of all 2 x 2 matrices of the form in (1) as follows. Given a complex number z = a + bi , let T(2) - [; )- a a (a) Suppose that zi and z2 are complex numbers and c1 and c2 are real numbers. Show that T(c1č1 + c2=2) = c1T(z1) +c2T(z2). (b) Show that T(2122) = T(21)T(22) for all complex numbers z1 and z2. Note the complex multiplication on the left in contrast with the matrix multiplication on the right. (c) Prove that if z is an arbitrary nonzero complex num- ber, then T(2-') = {T(2)}"". where z- = 1/z and {T(2)}¬' is the inverse of the matrix T(z).
1. Motivated by Example 2, let us define a function or trans- formation T from the set of all complex numbers to the space of all 2 x 2 matrices of the form in (1) as follows. Given a complex number z = a + bi , let T(2) - [; )- a a (a) Suppose that zi and z2 are complex numbers and c1 and c2 are real numbers. Show that T(c1č1 + c2=2) = c1T(z1) +c2T(z2). (b) Show that T(2122) = T(21)T(22) for all complex numbers z1 and z2. Note the complex multiplication on the left in contrast with the matrix multiplication on the right. (c) Prove that if z is an arbitrary nonzero complex num- ber, then T(2-') = {T(2)}"". where z- = 1/z and {T(2)}¬' is the inverse of the matrix T(z).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:31. Motivated by Example 2, let us define a function or trans-
formation T from the set of all complex numbers to the
space of all 2 x 2 matrices of the form in (1) as follows.
Given a complex number z = a + bi, let
a
b
(a) Suppose that z1 and z2 are complex numbers and c1
and cz are real numbers. Show that
T(c121 + c2z2) = cT(z1) +c2T(z2).
(b) Show that
T(z12) = T(z1)T(z2)
for all complex numbers z1 and z2. Note the complex
multiplication on the left in contrast with the matrix
multiplication on the right.
(c) Prove that if z is an arbitrary nonzero complex num-
ber, then
T(2-') = {T(2)}"'.
where z- = 1/z and {T(z)}~1 is the inverse of the
matrix T(z).
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