= 3. Throughout this exercise, i √1 € C. Recall that if z = a + bi € C is a complex number where a, b = R, then z = a - bi = C is called the complex conjugate of z. You may freely use the following facts about complex conjugation for any w, z = C: z+w=z+w, and that z = z if and only if z E R. Consider the following set of complex 2 × 2 matrices: {| : w, z e c}. ІНІ = Finally, put B = {1, i, j, k}, where 1 = Z -W W Z 0 ·[1], ¹69] " zw=z.w, j= [J], K=[ 3]. k= (a) Verify that B is a subset of H and that the following equations holds: i² = j² = k² = ijk = -1. (b) Show that H is closed under matrix addition and scalar multiplication by real numbers. Is it closed under scalar multiplication by complex numbers? (In particular, this shows H is a subspace of the vector space of 2 × 2 complex matrices viewed as a real vector space.) (c) Prove that B is a basis for H, regarded as a real vector space. What is the dimension of H? (d) Show that H is closed under matrix multiplication. (e) Check that if A E H and A is not the zero matrix, then A is invertible and A-¹ € H. (f) Does H form a field under the operations of matrix addition and matrix multipli- cation? Give reasoning for your answer. (In fact, H is a famous arithmetic discovered by William Rowan Hamilton, known as the quaternions. The quaternions have important applications in computer graphics and other fields. The equations in (a) became famous after he carved them in stone on Broom Bridge in Dublin in 1843.)
= 3. Throughout this exercise, i √1 € C. Recall that if z = a + bi € C is a complex number where a, b = R, then z = a - bi = C is called the complex conjugate of z. You may freely use the following facts about complex conjugation for any w, z = C: z+w=z+w, and that z = z if and only if z E R. Consider the following set of complex 2 × 2 matrices: {| : w, z e c}. ІНІ = Finally, put B = {1, i, j, k}, where 1 = Z -W W Z 0 ·[1], ¹69] " zw=z.w, j= [J], K=[ 3]. k= (a) Verify that B is a subset of H and that the following equations holds: i² = j² = k² = ijk = -1. (b) Show that H is closed under matrix addition and scalar multiplication by real numbers. Is it closed under scalar multiplication by complex numbers? (In particular, this shows H is a subspace of the vector space of 2 × 2 complex matrices viewed as a real vector space.) (c) Prove that B is a basis for H, regarded as a real vector space. What is the dimension of H? (d) Show that H is closed under matrix multiplication. (e) Check that if A E H and A is not the zero matrix, then A is invertible and A-¹ € H. (f) Does H form a field under the operations of matrix addition and matrix multipli- cation? Give reasoning for your answer. (In fact, H is a famous arithmetic discovered by William Rowan Hamilton, known as the quaternions. The quaternions have important applications in computer graphics and other fields. The equations in (a) became famous after he carved them in stone on Broom Bridge in Dublin in 1843.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please do not provide solution in image formkat thank you!
![3. Throughout this exercise, i = √1 € C. Recall that if z = a + bi € C is a complex
number where a, b = R, then z = a - bi Є C is called the complex conjugate of z. You
may freely use the following facts about complex conjugation for any w, z € C:
z+w=
ІНІ
z + w,
and that z = z if and only if z € R. Consider the following set of complex 2 × 2 matrices:
2
H-{[*]
w, z =c}.
Finally, put B = {1, i, j, k}, where
Z
zw = z.w,
¹-6 -6 9 -3, - d
1
=
=
2
j
(a) Verify that B is a subset of HI and that the following equations holds:
¡² = j² = k² = ijk = -1.
(b) Show that H is closed under matrix addition and scalar multiplication by real
numbers. Is it closed under scalar multiplication by complex numbers?
(In particular, this shows H is a subspace of the vector space of 2 × 2 complex
matrices viewed as a real vector space.)
(c) Prove that B is a basis for H, regarded as a real vector space. What is the dimension
of H?
(d) Show that H is closed under matrix multiplication.
(e) Check that if A E H and A is not the zero matrix, then A is invertible and A-¹ € H.
(f) Does H form a field under the operations of matrix addition and matrix multipli-
cation? Give reasoning for your answer.
(In fact, H is a famous arithmetic discovered by William Rowan Hamilton, known as
the quaternions. The quaternions have important applications in computer graphics and
other fields. The equations in (a) became famous after he carved them in stone on Broom
Bridge in Dublin in 1843.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b4ab23-f6d5-4206-a94e-63cad8267fd4%2Ffa1bb4bc-c107-47d2-b882-bf08c87d7c86%2F5frl11_processed.png&w=3840&q=75)
Transcribed Image Text:3. Throughout this exercise, i = √1 € C. Recall that if z = a + bi € C is a complex
number where a, b = R, then z = a - bi Є C is called the complex conjugate of z. You
may freely use the following facts about complex conjugation for any w, z € C:
z+w=
ІНІ
z + w,
and that z = z if and only if z € R. Consider the following set of complex 2 × 2 matrices:
2
H-{[*]
w, z =c}.
Finally, put B = {1, i, j, k}, where
Z
zw = z.w,
¹-6 -6 9 -3, - d
1
=
=
2
j
(a) Verify that B is a subset of HI and that the following equations holds:
¡² = j² = k² = ijk = -1.
(b) Show that H is closed under matrix addition and scalar multiplication by real
numbers. Is it closed under scalar multiplication by complex numbers?
(In particular, this shows H is a subspace of the vector space of 2 × 2 complex
matrices viewed as a real vector space.)
(c) Prove that B is a basis for H, regarded as a real vector space. What is the dimension
of H?
(d) Show that H is closed under matrix multiplication.
(e) Check that if A E H and A is not the zero matrix, then A is invertible and A-¹ € H.
(f) Does H form a field under the operations of matrix addition and matrix multipli-
cation? Give reasoning for your answer.
(In fact, H is a famous arithmetic discovered by William Rowan Hamilton, known as
the quaternions. The quaternions have important applications in computer graphics and
other fields. The equations in (a) became famous after he carved them in stone on Broom
Bridge in Dublin in 1843.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Introduction
VIEWStep 2: a) To prove (I, i, j, k} is a subset of H
VIEWStep 3: b) To prove that H is closed under matrix addition and scalar multiplication
VIEWStep 4: c) Find the order of H
VIEWStep 5: d) To prove H is closed under matrix multiplication
VIEWStep 6: e) To prove that the inverse of A exist in H
VIEWStep 7: f) To check whether H is a field
VIEWSolution
VIEW
Step by step
Solved in 8 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
