Question

3. Determine whether the function (p,q)= a, b + ab + ab, where p(x) = 4, + ax + ax and g(x)=bo bat checking the following axioms of the definition of an inner product: 40 72x+ax and g(x)=b, +bx+b.rbelong to P2 represents an inner produd A) (5,8)=(8,8) B) (f, g+h)=(5,8)+(f,h) C) c(f,)=(cf,8) for any scalar c D) (f, f)20, and (f, f)=0 if and only if f(x)=0

linear algebra

Answer 1

@ het f(x) = 4. + 4x + 4,42 911) = bo + b, x + b, x2 Then <+18> = Gobo +9,61 + 02 be - boa. + b, a, + b2 92 = <g,f> ® Let fwa 9. +9, x + 22x2 8 (1) = bo+b, x + b₂ n² hin = 6 + ( x + C₂ H2 Then g(u)than) = (bo+co) + 16,+G) x + (6 +62) 22 and <figth> = Golbo +Co) + 9, 16,+G) + 162+6) = (4. bo +4, 6, +ą, b2) + (%. Co + 9,4 +0,42) = <fig>+ <fich> Scanned with CamScanner

© We have <<tig> = c( 4o ba +916,+ be] = (ca.) bo + (9.36, + (092) bz = <cf, g> © Let fin) = Q. +94 +0,x2, then 1'920 < f;f> = 92 +92 +92 to 19+20, 12 zo since 9² +9² +92 2 zo if and only if ao = 9,=92=0 therefore <f, f>=0 if and only if f(n) = Hence the function <b,q>= Go bot 9, bet az bz is an inner broduct. Scanned with CamScanner
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