Question

1 The infinite family {Pn(x)}^=o of Legendre polynomials Pn(x) forms a complete orthogonal family on the interval [-1, 1]. If we delete the first element Po(x) = 1 from the set, the remaining family {Pn(x)}=1 also forms a complete orthogonal set. 2 Let {Xn}n=1 be a complete orthogonal family of functions for the vector space L[0, 1]. Then enlarging the set by adding to this set the vector 2X5 + 3X18, we end up with a second infinite set {2X5 + 3X18, Xn}=1. This new set of vectors is also a complete orthogonal family.

Answer True or False and explain

Answer 1

1. Infinite family {Pa (2) } of legendre polynomials forms a complete orthogonal family on the interval This is true statement. Since, we know that for Pn (2) a Legendre polynomial Pm (2) In (7) def o if man (enti if m=n. - m20 Hence, for min (Pm (2) Pn (2) dx = 0 2. ffn (2.) 5 forms an orthogonal set. If we delete the first element Polyal from the set, the remaining family { Pn Golf also forms a complete orthogonal set This is a true statement. Since we know, s Pn (2) Pn (x) dx s f o, men ent, man It is also true for m, n71. Hence of incesto forms an orthogon un )

& be a complete orthogonal family of functions for the vector space Lo, 1] Then adding 2x5 + 3X18" we get a second infinite set, 2XG + 3X18, Xn ] This new set of vectors is also orthogonal . As If we define Standard inner product then, Since {xn as orthogonal. 150, for itš. {X?, x>= 0 & 1j. Now, take it 5, 18. 2x5+ 3X18, X, =2X5, Xo>+ 3 LX18, X) = 2.0 +3.0 = 0. forms a complete 218 2x5 +3X18, XnI o 0118, Ah Jual orthogonal set.