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MATH 4140/7140 Homework 5 Due 7/21/2016 Do all problems on different paper. I will not grade anything written on this page. Show
your work. Answers without work will be given little or no credit.
1. (a) Show that for any v, w in an inner product space V , the following identity holds:
|| v + w ||2 + || v − w ||2 = 2|| v ||2 + 2|| w ||2
(We know that an inner product space is always a normed linear space as a norm
can be defined from the inner product. However, not every normed linear space is
an inner product space, meaning the norm may not arise from an inner product.
In fact, a normed linear space is an inner product space if and only if the norm
satisfies the above equality for every v, w ∈ V .)
(b) Show that the infinity norm || · ||∞ on R2 defined by ||(x1 , x2 )T ||∞ = max{|x1 |, |x2 |}
is not derived from an inner product.
2. Consider the vector space C[−1, 1] with the standard inner product, that is,
Z 1
f (x)g(x) dx.
hf, gi =
0 Find an orthonormal basis for the subspace spanned by 1, x, x2 .
3. Consider the following orthonormal basis for R3 (you do not have to show it is an
orthonormal basis):

u1 = 1
1
1
√ , √ , √
3
3
3 T 
, u2 = 2
1
−3
√ , √ , √
14
14
14 T 
, u3 = 4
−5
1
√ , √ , √
42
42
42 T
. Use the inner product to write v = (−1, 1, 2)T as a linear combination of these basis
vectors. Then find the length of v using Parseval’s formula.
4. Recall that if V is a vector space and ϕ : V → V is a linear operator, then we say that a
nonzero vector v ∈ V is an eigenvector of V if ϕ(v) = λ v for some scalar λ, when this
happens we say that λ is an eigenvalue of ϕ.
(a) Find the eigenvalues and corresponding eigenvectors of −1 0 1
A = −3 4 1
0 0 2 1 (b) Let V = R∞ = {(x1 , x2 , ...., xn , ...)T | xi ∈ R}. Notice that R∞ is a vector space
with addition defined by component-wise addition and scalar multiplication defined
by α · (x1 , x2 , ...., xn , ...)T = (αx1 , αx2 , ...., αxn , ...)T . Consider the linear operator
ϕ : R∞ → R∞ defined by ϕ((x1 , x2 , ...., xn , ...)T ) = (0, x1 , x2 , ...., xn , ...)T . (You do
not need to show that R∞ is a vector space nor do you need to show ϕ is a linear
operator). Show that Ï• does not have an eigenvector.
(c) Consider the differential operator D : C ∞ (−∞, ∞) → C ∞ (−∞, ∞) defined by
D(f (x)) = f 0 (x). Show that every λ ∈ R is an eigenvalue of D. (Hint: Consider a
function that is closely related to its derivative.) 2

Attachments:

• 1492746517-HW 5.pdf