Question

17. ({(x, kx) x any real, k constant), coordinate-wise addition) 8. ({ f(x) 105x31}, +) 9. ({e* x any real}, :) 10. (P2 = { ax? + bx +ca,b,c any real}, +) 11. ({In x | x>0}, +) - naordinate-wise addition)

please solve 7,8,10,11

find property of vector like closure , associative all 5 list is on that picture with explanation

Answer 1

Il x any real, k constant 3 coordinate-wise codelition) 1) CLOSORE : _ het (a, kx) , (y, hy) Ev , where x, y ElR then 2, fax)+ (Y, Hy) = (x+y, k(x+y)) EV ao ary ER XYER 2) ASSOCIATIVE let (a, kx) EV (Y, ky) EV, (3, ks) Ev , where hy, at R (X,kx) + ((%i ky ) + (3,ka)) = (x,ku) + (343) (t+a)) = (x+4+3 , *x+k(978)) = (atyta, k (x+y+a)) Also, ((2, ku) + , kt))+ (3,ka) = (x +4, k (+) + (.ka) = (x+4, 8] +(2+413)) thus 2 db) +(1. 4) + (8 ) = (.da) (tky) + 6.ka).

3. TOENTITY: Let (ac, kx) eV then let u= (ui, ku) EV, U, ER such that (oc, kx) + 4 = (x4kx) =) (2, 4x) + (4., ku) = (x, tx) a) [3+4,, k(x+4)) = (x, kv) - equating components in both sides we get 3) 3+4 = x. , ke +4.) = * * * 4,20 sore his internet 7) U = (4.,kus) = (0,0) thus, (o, ko) e Vio identity in v such that (2,6x) + (0,ko) = (2,6x) 4. Inverses. Let (@,kw) EV and 3 w = (w, +kw) GV, WIE IR such that (3, kx) + (bw,+kws) = (0,40) + (x+w, , 6(x+w)) = 0,40) 1)

thus (-x, (-)) is inverse of (2,4z) in V. thus for each (2c, kx) in V, there is a (-x, k(-x) in v such that (1,43)+ (-3, (-)) . ( 4 ) and 6-2, kl»»)) + (2,4x) = (0,6.0) where (s, ko) in identity in V. 5. COMMUTATIVE : let (2, kr) , (Y,ky) EV; Whore X,Y E IR then (2, da) + (Hity) = (aty, k(x+y)) = (x+2, k (4+*)) : 8 k ) + ( AA) thus t (x, kx), (y, ky) EV (236x) + (Hikay) - Chky) +(2,4x)