Question

The function, By justifying your answer, determine whether 7 defines an loner Product cas (cui nas Ws, un, (wu. Var Vauva) ) - Li. Wy - 5w, vg and Va " (b) Kuus), u. Nad >=V7 Live + us to and V=R*

Answer 1

solution @ v=IRY T(4,, U2, U3, (4), (V, V2, V3 V4 ) = u V4-542.43 It is not an innerproduct en DR4 Becaure. for (1,1,0,0), co, o, 1, 1) E IRS KC!,!,0,0), (0o,1,1)>=1-5=-4 4 < 60,0,1,1), (1,1,0,0)) = 0-5.0 = {(1, 1,90), (0,0,1,1)) <60,0,1,1),(1,1,0,0)) =) <a,y) £ {9, 2) for x=(1,1,0,0), Y = (0,0,1,1) Hence it is not an inner product G pace

V = TR² + < (U,,U₂), CV,,V2)) = 2 u,u,+ U V inner product on v. go for x=(4,,,) y = CV., Y) E DR² 4 w=(wi, W2) ELR² 0 Tax, y) = {(4, Me), CV., V.)) - 12 urit U2N2 =J2 vu+ V₂ U = < (V., V2), (4), U2)) [7 ,x) ♡ (x, ytw) =< (4,42), (V, two vzt W2)) - sau, (utwt U CV t₂) = zuiv, tJ2u, w, tu2 t u uz (zuu, + U₂ V2)+(J2 u, w, tchuh) <(4,4), (V1, V2)) + Cu, 42), CW,,uz)) - <a, y + <x, w) - <a, ytw) = <a, y) + <x, w) KER k x, y) = K(Cu, k), (V., Vz)) = K[r2ui vit U2 V₂] - 2 (ku,) vit (K U) ₂ LOkul, K2), (v., V₂)) <kx, y) = Kla,y) = <kx,y) 6 Laa) = < (41,42), (ui, U2)) = 2 u² tus 70 CU,, UG [R = 42, 4270) O

> <cu, 4), CU,,U2)) = 0 2 urt=0 Now, J2 4² 4² as u,, U ER J2 u²t us -0 =) u, = U2 = 0 Ću, ₂) = cool a-o 26. 20.0% 0 then Hence all five properties satisfied, so it is inner product