Question

Consider the linear transformation T: R4 + R2 defined as T(11,12,13,14)=(-221 +22 +2 14,322 -14) Find the standard matrix for T: ab sin(a) f 8 a 12 . ar What is the dimension of ker(T)? Number Is T one-to-one? Enter one yes no Write the standard matrix for HoT, where H is the reflection of R2 about the line y=1. ab sin(a) f αο α Ω TI д

Answer 1

» R2 for R2 Now Standard basis for R4 soln :- Given linear transformation T: R4 defined by *T (X, , Xs, Qu) = (-2.861+ 2+ adly-3X2-w) Stendard basis for e = (1,0) & C2 = (0.1) Consider any (ab) E R² st (a,b) Callio) + (0.1) W, = (1,0,0.9).V2-0190), V3=6019) vy (0,0,0,1) By TUI) = 62,0) = -2 (1,0) +010,1) = (1, -3) - 1 (1,0) +3701) t(U3) = (0,0) 0.(1,0) +00,1) T(vy) (2,-) 2 (leo) + Cu @.1) Hence mateix for T 2 1 & E T (12) is 2 1 D# 0 -3 o -) Scanned with CamScanner

To determine kest) -2x + x + 2xy =0 - 3 H = xy =0 Here two equation & 4 Unknown are free variable = number of Xz 4 xy o dim keilt) free variable Hence dimke (V) - 2 NOW T is one-one if and only if dim ket) = 0 But dém kercs ) = 2 70 Hence T is not one-one. Ans: NO CS Scanned with CamScanner