Question

1 1 0 -1 Exercise 2. Let A = 0 1 0 in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R”, set g(ū) = WT AV E R. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]c,b in the bases C = {1} and B = { 9 8 B |}? (ii) Let f: R3 + R be the function defined by f(w) = vſ Aw E R. Show that f(ū) = g(ū) for every W E R3 (iii) Find a vector v such that g(w) v? w for every û ER3. =

Answer 1

Sol 2i To show gi R² R is a linear transform Ginen : g(W) a WTAW let 13 w 11 e R² وفيا وفيا Then g (w) [W, We wg 1 :10] - -1 O =[w, We Wg] "T 13 Then That is a zero ot let gew) & N ER² is transform w we ER? glu,+ wa) geun) & gewe) Now then let XEIR WER? gan) & glu) Thus g is linear transform

f(3) = gew) & W GIRI Now will find the matrix [g JC, B & we R² since g cão) = 0 Oxl 11 o o Ох 1 [!]) 3([:]) g ( []) [glere Ox! 0 [ooo] Thus B Solli To show Ginen: flwga ytan رقام = Let ونا ولها f(W) [10] = [io 1] [w, Wz W2 -WitW, Wi-Wg -WitWz Thus f (w)=0 & N E IR?

& W ERI f (W) = gruno Hence SOL ) vector such that glu) = the find every for W ER3 tet 13 Eva If gewo) a stw, then g{ ]) = v[] *7[:] >> o = [V, V, Vs ] ñ O Also a(:D) - To - O = this vs vs?[:] 12 V2 And 0+1(8]) = ** [:) = [v, To ] vs 13 O Thus 13 *WER? 1 [ such that gewo) = vTwo