parts a-d. Thanks, this is linear algebra
- Hi Could You Please Just Do Part E I Believe You Will Need Parts A D Thanks This Is Linear Algebra 1 (55.42 KiB) Viewed 26 times
Hi could you please just do part e)? I believe you will need parts a-d. Thanks, this is linear algebra
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Hi could you please just do part e)? I believe you will need parts a-d. Thanks, this is linear algebra
Hi could you please just do part e)? I believe you will need
parts a-d. Thanks, this is linear algebra
1. In this question, you will be using the following trigonometric identities: cos²a + sin² a = 1 (1) cos(a + 3) (2) = cos ax cos 3-sina sin = sin a cos 3 + cos a sin 3 sin(a + 3) (3) where a. 3 € R. You do not need to prove these identities. You may also use without proof the fact that the set sin a is exactly the set of unit vectors in R². Now for any real number a. define Ra cos a-sina sin a cos a (a) Prove that for all a. 8 € R. R₂R₁ = Ra+ (b) Using part (a), or otherwise, prove that R, is invertible and that R¹ = R., for all a € R. (e) Prove that for all a € R and all x. y € R². (Rax). (Ray)=x-y (d) Suppose A is a 2 x 2 matrix such that for all x. y € R², (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 x 2 matrix. (i) Show that there are real numbers and a such that <= 1111 [cos a] sin a (ii) Let a € R. Use the invertibility of Ra to prove that there are unique 112,22 ER such that sin 4 [cos a sin a = 12 +4422 Cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R₂U for some a € R and some upper-triangular matrix U. (iv) Suppose that B = R₁U = R₂V, where a. ß ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V. =
parts a-d. Thanks, this is linear algebra
1. In this question, you will be using the following trigonometric identities: cos²a + sin² a = 1 (1) cos(a + 3) (2) = cos ax cos 3-sina sin = sin a cos 3 + cos a sin 3 sin(a + 3) (3) where a. 3 € R. You do not need to prove these identities. You may also use without proof the fact that the set sin a is exactly the set of unit vectors in R². Now for any real number a. define Ra cos a-sina sin a cos a (a) Prove that for all a. 8 € R. R₂R₁ = Ra+ (b) Using part (a), or otherwise, prove that R, is invertible and that R¹ = R., for all a € R. (e) Prove that for all a € R and all x. y € R². (Rax). (Ray)=x-y (d) Suppose A is a 2 x 2 matrix such that for all x. y € R², (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 x 2 matrix. (i) Show that there are real numbers and a such that <= 1111 [cos a] sin a (ii) Let a € R. Use the invertibility of Ra to prove that there are unique 112,22 ER such that sin 4 [cos a sin a = 12 +4422 Cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R₂U for some a € R and some upper-triangular matrix U. (iv) Suppose that B = R₁U = R₂V, where a. ß ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V. =
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