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Let B = be any 2 x 2 matrix. COS a (i) Show that there are real numbers un and a such that u11 sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique u12, U22 € R such that [cos a] + u22 = u12 [sin a (iii) Use parts (i) and (ii) to show that B can be expressed in the form - sin a] cos a B = RU for some a €R and some upper-triangular matrix U. (iv) Suppose that B = RU = RV, where a, B ER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.

Let B = be any 2 x 2 matrix. COS a (i) Show that there are real numbers un and a such that u11 sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique u12, U22 € R such that [cos a] + u22 = u12 [sin a (iii) Use parts (i) and (ii) to show that B can be expressed in the form - sin a] cos a B = RU for some a €R and some upper-triangular matrix U. (iv) Suppose that B = RU = RV, where a, B ER 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. COS a (i) Show that there are real numbers un and a such that u11 sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for un in terms of a and c. (ii) Let a € R. Use the invertibility of R, to prove that there are unique u12, U22 €R such that cos a = u12 - sin a + u22 cos a sin a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a €R and some upper-triangular matrix U. (iv) Suppose that B = R,U = R3V, where a, B eR and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
Transcribed Image Text:(e) Let B = be any 2 x 2 matrix. COS a (i) Show that there are real numbers un and a such that u11 sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for un in terms of a and c. (ii) Let a € R. Use the invertibility of R, to prove that there are unique u12, U22 €R such that cos a = u12 - sin a + u22 cos a sin a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a €R and some upper-triangular matrix U. (iv) Suppose that B = R,U = R3V, where a, B eR and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
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