Question

For real non-zero constants a and b, consider the linear transformation T: R3 + R3 defined by orthogonal reflection in the plane ay + b2 = 0 where orthogonality is defined with respect to the dot product on R3 x R3. Find in terms of a and b the numerical entries of the matrix Aſ that represents T with respect to the standard ordered basis {el, C2, C3} of R3.

Answer 1

To find the AT we matrix that represents T wor.t the ordered basis Just need to find Tie,), Ten) Trez) Note that, e, E T. so, its reflection with respect to TT is itselt. =) Te, e.

Now and perpenpendicate any straight line through e, to T is given by, r(t) = {e. + t. (o, a, b } TER}. (: (0,a, b) is the direction ratio of the normal vector to TU) 1 Note that, + 6 18 : Te Er(t). => Te, = e+ + (0,2, b) for some Further, Il Te, -e, 11 = 2x dength of perpendiculas Projected on T from e. → 11 t (0,4,5) 11 2x a. + 6.0 q+62 ant 2a mertua -> trattbt ( Il denotes the nor induced by dot product in t 20 4+5 R?) Hence 2a arty طم2 2a² a+b? ) 9²+5² cab 275 32 9762 Te, = et t. Coà,5). (0,1,0) + (09,b) ( = (0, Battle /gab atbr et Zalo ез Similarly, to find Tez E Vict) {est 7, (0,9,b) : t, EIR}. - Te = ez + t, Co, a, b) for some t, Ero "Tez-e311 =24 length of perpendicular I from > lt, (, a,b) || = 2 * aio + bil ratst t, a q+62 we proceed that and projected 2b Hence Tez: ezt 2,1)= 0, Lab 97² 1 26² 975 ab 979 9²767 ez

AT Hence 2 Crab Ans) 0 3a752) a²+62 zab atb 735 atb?