Question

(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T be the transformation that reflects R2 over the line y = x. That is, T (| Guess what det (T) should be, then check by finding the matrix for T and computing its determinant. (c) Rotation matrices in R3 are more complicated than in R2 because you have to specify an axis of rotation, which could be any line through the origin. If T is a linear transformation describing a rotation in R3, what would you expect det (T) to be? Now let's check your expectations. The three basic 3D rotation matrices are given as follows, representing counter-clockwise rotation by an angle a about the x, y, and z axis respectively. Compute det(A) for each one, and compare it to what you expected the determinant to be. [1 0 0 1 [cos(0) 0 - sin(0)] [cos(0) - sin(0) 07 0 cos(0) – sin(0) 0 1 0 sin(0) cos(0) 0 sin(o) cos(0) ] (sin(0) O cos(0) ] [ 0 0 1] Note: In fact, any rotation matrix in 3 dimensions can be written as a composition of these basic matrices! Look up the "basic 3D rotation matrices" on Wikipedia if you want to learn more. (d) Let T: R3 → R3 be projection onto the xy-plane, so T = Y. What is det(T)? Guess first and then check using a matrix.

Answer 1

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transformation (6 same question as a) only this time let T be the that reflects R² over the lin yaz. That is, t/ biless what det (1) should be then check by finding the matrix fort and computing its delominant. that reflects &? over the I be the transformation linear Live 928; 10 line Yzx; he T 1 ) standard onderd guess ? det (t) = 5 To calculate matrix for T, cue consider the basis {(1,0), (01) 3 of R? (9))(9) 20 (943.(1). ((?) - () -2.(1) 70 (1) Hence matrix of t a TO det (T) 20 (1) -1 - checked.

(C) Rotation matrices in R3 are complicated than in R² because You have to specify an axis of rotation, which could be any line through the origin. if t is a linear transformation a rotation describing in R² what would you expect det (1) to be a natation counter clockwise W Tx Ty, Tz denote the 2 axis axis.yaxis respectively by an angle o, about x Messa del (T -det (Ty)= det (Tz)=1 from wikipedia, we get, o To caso sino -sine caso I cosa o sino] Tч , І L-sind o caso cond sino -sino coso of o det (TX) Costo (sin 20) 1 sino COSO

Cos o sinol det (1₂) 1. / ? COS 1 be (d let TR²R projection onto the xy -plane sotly- a matrix. what is det (T)? Guess first and then check using TR3 R be the projection onto the xy Plane. Guess -det (T)-a (3-12. buves - dd (1): comesponding to TT, we are considering To calculate the matrix the standard ordered basis a 0,0,0), (0.7.0), (0.0, 1) 3. * * 0 13:13

09-11--1) 13 [tool :: Toto Hence dd (1) 1300 Totol.o. - checked.