[a b] (e) Let B be any 2 x 2 matrix. d [cos a] sin a = U11 (i) Show that there are real numbers u11 and a such that a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a E R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that CO a sin a И12 + U22 sin a COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R.U for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RQU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
[a b] (e) Let B be any 2 x 2 matrix. d [cos a] sin a = U11 (i) Show that there are real numbers u11 and a such that a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a E R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that CO a sin a И12 + U22 sin a COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R.U for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RQU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
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
Only need the answer for (e)(iv)
Transcribed Image Text:- sin a
[cos a
Ra
COS a
sin a
COS a
![a b
(e) Let B
be any 2 x 2 matrix.
c d
а
COS a
U11
(i) Show that there are real numbers u11 and a such that
sin a
Hint: express
as a scalar multiple of a unit vector, and hence find an
expression for u11 in terms of a and c.
(ii) Let a e R. Use the invertibility of Ra to prove that there are unique
U12, U22 E R such that
sin a
[b]
= U12
cos a
+ U22
sin a
COS a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = RaU
for some a E R and some upper-triangular matrix U.
(iv) Suppose that B
triangular. Prove that if B is invertible, then U =±V.
RaU = R3V, where a, ß E R and U and V are upper-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbadb61e7-6232-4e30-8241-a2efa7d79abe%2F02c2c41c-0a00-4223-add3-9cd2266b81ba%2Fke85xqe_processed.png&w=3840&q=75)
Transcribed Image Text:a b
(e) Let B
be any 2 x 2 matrix.
c d
а
COS a
U11
(i) Show that there are real numbers u11 and a such that
sin a
Hint: express
as a scalar multiple of a unit vector, and hence find an
expression for u11 in terms of a and c.
(ii) Let a e R. Use the invertibility of Ra to prove that there are unique
U12, U22 E R such that
sin a
[b]
= U12
cos a
+ U22
sin a
COS a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = RaU
for some a E R and some upper-triangular matrix U.
(iv) Suppose that B
triangular. Prove that if B is invertible, then U =±V.
RaU = R3V, where a, ß E R and U and V are upper-
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
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,
