Extra Credit Problems
Recall that
C[a, b] = {f : f is continuous on [a, b]}.
Most of the functions we are familiar with from Calculus and Algebra are in C[a, b],
such as polynomials, exponentials, and sine and cosine. In Chapter 4 of the text, we
noted that C[a, b] is a linear space. In fact, C[a, b] has some even nicer structure. It
is actually an inner product space with inner product
Z b
f (x)g(x) dx.
hf, gi =
a
For the definition of an inner product space and the properties of an inner product,
see section 5.5 of your text. We are already familiar with one example of an inner
product space. The dot product in Rn is an example of an inner product. This
means that all the formulas that we have derived and theorems that we have proved
that involve the dot product in Rn still hold in any inner product space if the dot
product is replaced by the inner product for the space in which we are interested.
The purpose of this exercise is to apply some of the experience we have with Rn as
an inner product space to the more complicated space C[0, 1].
Problem : Consider the function sin(6πx) ∈ C[0, 1]. Find the function from P5 ⊂
C[0, 1] that is closest to sin(6πx) with respect to the norm induced by the inner
product above. This can be done in two steps:
(a) The standard basis for P5 is {1, t, t2 , t3 , t4 , t5 }. Use this basis as a starting point
to produce an orthonormal basis.
(b) We showed in section 5.4 that the closest point to a given element f of a linear
space from a subspace V is projV (f ). Let V = P5 and f = sin(6πx) and compute
projV (f ).
Show all your work on parts (a) and (b) and provide a graph of the solution along
with sin(6πx). Each part will be graded out of 5 points. Also let me know to which
portion of your grade you would like the extra credit applied (Quizzes, Tests, Final,
or Projects).
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