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а (e) Let B = be any 2 x 2 matrix. COS a [sin a (i) Show that there are real numbers U11 and a such that = U11 а 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 U12 + 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 = RaU = R&V, where a, ß E R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.

а (e) Let B = be any 2 x 2 matrix. COS a [sin a (i) Show that there are real numbers U11 and a such that = U11 а 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 U12 + 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 = RaU = R&V, where a, ß E R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.

Advanced Engineering Mathematics
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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(e) Let B be any 2 x 2 matrix. а COS A (i) Show that there are real numbers U11 and a such that U11 sin a 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 = 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-
Transcribed Image Text:(e) Let B be any 2 x 2 matrix. а COS A (i) Show that there are real numbers U11 and a such that U11 sin a 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 = 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-
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