Let fa,fb,fc be theLagrange polinomials associated with distinct real numbers a,b,c respectively.Define T:P2(R?

[970]

Define T2® -- P2® by T(g) = g(a)fa + g(b)fb.
a)Show that T is a linear operator
b)Expalin whether or not T is a projection.
c)Find [T}B, where B = {fa,fb,fc}

[KB]

a) For any polynomials g and h, and scalars c, d:
T(cg + dh) = (cg + dh)(a) fa + (cg + dh)(b) fb
................= (cg(a) + dh(a)) fa + (cg(b) + dh(b)) fb
................= cg(a) fa + cg(b) fb + dh(a) fa + dh(b) fb
................= c (g(a) fa + g(b) fb) + d (h(a) fa + h(b) fb)
................= c T(g) + d T(h).

Hence, T is linear.
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b) Check whether T^2 = T.
T^2(g) = T(T(g))
.........= T(g(a) fa + g(b) fb)
.........= (g(a) fa + g(b) fb)(a) fa + (g(a) fa + g(b) fb)(b) fb
.........= [(g(a) fa(a) + g(b) fb(a)] fa + [(g(a) fa(b) + g(b) fb(b)] fb.

Since T^2(g) is not equal to T(g) in general, T is not a projection.
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c) Evaluate T at each element of B (in the presented order):

T(fa) = fa(a) fa + fa(b) fb + 0 fc
T(fb) = fb(a) fa + fb(b) fb + 0 fc
T(fc) = fc(a) fa + fc(b) fb + 0 fc

Hence, the matrix is
[fa(a) fa(b) fc(a)]
[fa(b) fb(b) fc(b)]
[...0.....0.......0..].

I hope this helps!