- Cosa Sina 1 In This Question You Will Be Using The Following Trigonometric Identities Cos A Sina 1 Cos A 3 Co6 1 (38.59 KiB) Viewed 17 times
- Cosa Sina 1 In This Question You Will Be Using The Following Trigonometric Identities Cos A Sina 1 Cos A 3 Co6 2 (38.59 KiB) Viewed 17 times
cos? a + sin? a
cos(a + B)
sin(a + B)
cos a cos 3 - sin a sin B
sin a cos 3 + cos a sin 3
(1)
(2)
(3)
where a, B € R. You do not need to prove these identities. You may also use without
proof the fact that the set
(cosa
:nER)
is exactly the set of unit vectors in IR?.
Now for any real number a, define
Ra=
caina
(a) Prove that for all a, B € R,
RaRe = Ratß
(b) Using part (a), or otherwise, prove that Ra is invertible and that Ra' = R_a, for
all a ER
(c) Prove that for all a € R and all x, y € IR?,
(Rax) • (Ray) = x•y
(d) Suppose A is a 2 x 2 matrix such that for all x, y € IR?,
(Ax) • (Ay) = x•y
Must it be true that A
Ra, for some a € R? Either prove this, or give a
counterexample (including justification).
(e) Let B
" be any 2 x2 matrix.
0) show that there are real numbers us and a such that d
= um (sma)
Hint copness " as a scalar multiple of a unit vector, and hence find an
expression for u11 in terms of a and c.
(i) Let a € R. Use the invertibility of Ra to prove that there are unique
212, 422 € R such that
(iii)
Use parts (i) and (ii) to show that B can be expressed in the form
B= RaU
for some a € R and some upper-triangular matrix U.
(iv) Suppose that B
= RaU
= ReV, where a, B € R and U and V are upper-
triangular. Prove that if B is invertible, then U = tV.
cosa sina 1. In this question, you will be using the following trigonometric identities: cos?a+sina = 1 cos(a + 3) = CO60.00 3-sina sin (2) sin(a +3) sino cos 8 + cosa sin (3) there 0.8 E R. You do not need to prove these identities. You may also use without proof the fact that the set sina is exactly the set of unit vectors in R. Now for any real number a define [cosa-sina R. COS (a) Prove that for all a, 3 ER RR=R (b) Using part (a), or otherwise, prove that R, is invertible and that R' = R. for all a E R (C) Prove that for all a € R and all x.y ER. (R.*)-(R_y) = x y (d) Suppose A is a 2 x 2 matrix such that for all XyER. (Ax). (Ay) = xy Must it be true that A - R. for some a € R? Either prove this, or give a counterexample (including justification) (e) Let B- b] [: 2] be any 2 x 2 matrix Toosa sina (1) Show that there are real numbers to and a such that Hint: erpress El as a scalar multiple of a unit vector, and hence find an Erpression for tu in terms of a and c. (ii) Let a E R. Use the invertibility of R. to prove that there are unique 12,33 € R such that Teosal - sinal 1112 sino ) Use parts (i) and (ii) to show that B can be expressed in the form B = RU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = R.U = RV, where a, 8 € R and U and V are upper- triangular. Prove that if B is invertible, then U = UV. + COS