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SWILA PROBLEM SET #9 SECTIONS 6.1-6.3 MONDAY, JULY 11

Goals for Problem Set: • Learn the definition of an inner product space. • Relate orthogonality with linear independence and norms. • Determine whether given bilinear functions define inner products. Instructions: The below problems are split up by difficulty level: Easiest, Mediumest, and Hardest. Within each difficulty level, the problems are ordered by how good I think they are to do. In other words, the problems that I think are best to do are listed earlier within each difficulty level. I recommend that you try problems from various sections or from sections that are new to you. Warnings: Labeling the difficulty of problems was performed rather imprecisely. Thus, some problems will be mislabeled. The same holds with ordering problems by how good they are to do. I also didn’t really proofread these problems so there may be typos, and I don’t know how to completely solve all of them.

I like the first 5 “Easiest” problems. I like the first 4 “Mediumest” problems. The first 3 “Hardest” problems look interesting.

Easiest Problems Exercise 1.0.1. (6.1)(Tao, pg. 51, # 1.4.9) Let H, H 0 be two complex inner product spaces. Define the direct sum H ⊕ H 0 of the two spaces to be the vector space H × H 0 with inner product h(x, x0 ), (y, y 0 )iH⊕H 0 := hx, yiH + hx0 , y 0 iH 0 . Show that H ⊕ H 0 is also an inner product space. Exercise 1.0.2. (6.1) Prove the Pythagorean Theorem: Let V be an inner product space and suppose x1 , . . . , xn ∈ V are pairwise orthogonal. Then n n X X 2 || xk || = ||xk ||2 . k=1

k=1

Hint: It isn’t too hard to do the case n = 2. Then use induction. Exercise 1.0.3. (6.1) We call a matrix A ∈ Mn (R) positive semidefinite, and write A ≥ 0, if hAx, xi ≥ 0 for all x ∈ Rn . Show that a diagonal matrix is positive semidefinite if and only if each of its entries is nonnegative. 1

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SECTIONS 6.1-6.3 MONDAY, JULY 11

Exercise 1.0.4. (Zhang, pg. 107, #5.4)(6.1) In this exercise, we will check if a certain function on Cn × Cn actually defines an inner product. Let x = (x1 , x2 , . . . , xn ) ∈ Cn and ||x||∞ = max{|x1 |, |x2 |, . . . , |xn |}. For x, y ∈ Cn , define hx, yi∞ = ||x||∞ ||y||∞ . Which of the following statements are satisfied for all x, y, z ∈ Cn , λ ∈ C? (a) (b) (c) (d) (e)

hx, xi∞ ≥ 0. hx, xi = 0 if and only if x = 0. hλx, yi∞ = λhx, yi∞ . hx, y + zi∞ = hx, yi∞ + hx, zi∞ . hx, yi∞ = hy, xi∞ .

Exercise 1.0.5. (6.1) (Zhang, pg. 116, #5.40) Let T be a linear transformation on an inner product space V . Let h·, ·i be an inner product on V . If one defines [x, y] = hT (x), T (y)i, what T will make [·, ·] an inner product for V ? Exercise 1.0.6. (6.3)(Petersen, pg. 175, #10) (Lagrange Interpolation) Select n + 1 distinct points t0 , . . . , tn ∈ C and consider hp(t), q(t)i :=

n X

p(ti )q(ti ).

i=0

(a) Show that this defines an inner product on Pn (C) but not on C[t]. (b) Consider (t − t1 )(t − t2 ) · · · (t − tn ) (t0 − t1 )(t0 − t2 ) · · · (t0 − tn ) (t − t0 )(t − t2 ) · · · (t − tn ) p1 (t) = (t1 − t0 )(t1 − t2 ) · · · (t1 − tn ) .. . p0 (t) =

pn (t) =

(t − t0 )(t − t1 ) · · · (t − tn−1 ) . (tn − t0 )(tn − t1 ) · · · (tn − tn−1 )

Show that pi (tj ) = δij and that p0 , . . . , pn form an orthonormal basis for Pn . (c) Given b0 , . . . , bn ∈ C, use p0 , . . . , pn to solve the problem of finding a polynomial p ∈ Pn such that p(ti ) = bi . (d) Let λ1 , · · · , λn ∈ C (they may not be distinct) and f : C → C a function. Show that there is a polynomial p(t) ∈ C[t] such that p(λ1 ) = f (λ1 ), . . . , p(λn ) = f (λn ). Exercise 1.0.7. (Zhang, pg. 113, #5.25)(6.1) Let V be an inner product space. (a) Show that ||x + y|| = ||x|| + ||y|| if and only if ||sx + ty|| = s||x|| + t||y||,

for all s, t ≥ 0.

(b) If ||x|| = ||y||, show that x + y and x − y are orthogonal. Explain this with a geometric graph.

SWILA PROBLEM SET #9

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Exercise 1.0.8. (6.1) Fix vectors x, y in an inner product space V with y 6= 0. Prove that we have equality in the Cauchy-Schwarz inequality (|hx, yi| = ||x|| · ||y||) if and only if x is a scalar multiple of y. Hint: Look at the proof of the Cauchy-Schwarz inequality. Exercise 1.0.9. (Zhang, pg. 107, #5.1)(6.1) Let V be a complex inner product space. Show 1 u is a unit vector, i.e., has norm 1, and that for any that for any nonzero vector u ∈ V , ||u|| v, w ∈ V , hv, hv, wiwi = |hv, wi|2 = hv, wihw, vi. Is it true that hhv, wiv, wi = |hv, wi|2 ? Exercise 1.0.10. (6.1) Let V be an inner product space. Recall that we say a subset S ⊆ V is orthogonal if and only if hx, yi = 0 for all x 6= y in S. Show that V is orthogonal if and only if V is the zero vector space. Exercise 1.0.11. (Zhang, pg. 59, #3.45)(6.2) Find the eigenvalues and corresponding eigenvectors of the matrix   2 1 0 A =  1 3 1 . 0 1 2 Show that the eigenvectors of distinct eigenvalues are orthogonal. We will later show with the Spectral Theorem that all symmetric bases have orthonormal bases of eigenvectors. Exercise 1.0.12. (6.3)(Polarization identities) Fix an inner product space V and x, y ∈ V . Prove the Polarization Identities: If F = R, 1 hx, yi = ||x + y||2 − ||x − y||2 . 4 If F = C, 4

1X j 1 hx, yi = i ||x + ij y||2 = ||x + y||2 − ||x − y||2 + i||x + iy||2 − i||x − iy||2 . 4 j=1 4 Exercise 1.0.13. (6.1) Let V be a complex inner product space. Show that for all v, w1 , w2 ∈ V and λ ∈ C, hv, w1 + w2 i = hv, w1 i + hv, w2 i. ¯ w1 i. hv, λw1 i = λhv,

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Mediumest Problems Exercise 1.0.14. (6.2) Let V be an inner product space. Prove that if nonzero vectors v1 , v2 , . . . , vr ∈ V are mutually orthogonal (hvi , vj i = 0 for all i 6= j), then {v1 , v2 , . . . , vr } is linearly independent. Hint: Pick off the coefficients of a linear combination equal to 0. Exercise 1.0.15. (Zhang, pg. 110, #5.15)(6.1) Let x = (x1 , x2 , . . . , xn ) ∈ Cn and ||x||∞ = max{|x1 |, |x2 |, . . . , |xn |}. If ||x + y||∞ = ||x||∞ + ||y||∞ , must x and y be linearly dependent? Is there an inner product h·, ·i on Cn so that hx, xi = ||x||2∞ for all x ∈ Cn ? Hint: Consider the parallelogram law for inner product spaces for the last part: ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2

for all x, y ∈ V.

Exercise 1.0.16. (6.3)(Petersen, pg. 174, #7) The goal of this exercise is to construct a dual basis of a basis x1 , . . . , xn for an inner product space V . We call {x∗1 , . . . , x∗n } ⊂ V a dual basis if hxi , x∗j i = δij . (a) Show that if x∗1 , . . . , x∗n exist then it is a basis for V . (b) Show that if x1 , . . . , xn is a basis, then we have an isomorphism T : V → Fn defined by   hx, x1 i   .. T (x) =  . . hx, xn i (c) Show that each basis has a unique dual basis (you have to show it exists and that there is only one such basis). (d) Show that a basis is orthonormal if and only if it is self-dual, i.e., it is its own dual basis. Hint: For part (b), note it suffices to show T is injective or surjective. Exercise 1.0.17.p(6.1)(Tao, pg. 50, #1.4.6) For an inner product space V and x ∈ V , recall we define ||x|| = hx, xi. (a) (Parallogram Law) For any inner product space V , prove the parallelogram law ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2 . (b) For each p ≥ 1, we define Lp (R) to be the set of all “nice enough” functions f : R → R R such that R |f |p < ∞. For example, piecewise continuous functions with bounded support lie in this space. This becomes a normed space with the norm Z 1/p p |f | ||f ||Lp (R) := . R

If p 6= 2, prove that this norm does not come from an inner product, i.e., there does not exist an inner product on Lp (R) such that ||f ||p = hf, f i (Hint: Consider functions of the form 1, χ[a,b) (x) := 0,

for all f ∈ Lp (R).

if a ≤ x < b otherwise.

for a < b.)


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Exercise 1.0.18. (Zhang, pg. 115, #5.31)(6.2) Let S = {u1 , u2 , . . . , up } be an orthogonal set of nonzero vectors in an n-dimensional inner product space V ; that is, hui , uj i = 0 if i 6= j. Let v1 , v2 , . . . , vq be vectors in V that are all orthogonal to S, namely, hvi , uj i = 0 for all i and j. If p + q > n, show that the vectors v1 , v2 , . . . , vq are linearly independent. Exercise 1.0.19. (6.1) Let (V, h·, ·i) be a complex inner product space. Is (V, [·, ·]) a real inner product space, where [x, y] := Rehx, yi? Exercise 1.0.20. (6.3)(Petersen, pg. 174, #6) Recall from Section 2.7 what it means to say that a matrix is in row echelon form. Given an arbitrary collection of vectors x1 , . . . , xm in an inner product space V , show that it is possible to find orthogonal vectors z1 , . . . , zn ∈ V such that [ x1 · · · xm ] = [ z1 · · · zn ]Aref , where Aref is an n × m matrix in row echelon form. Explain how this can be used to solve systems of the form   a1   [ x1 · · · xm ]  ...  = b. am Exercise 1.0.21. (Zhang, pg. 109, #5.11)(6.1) We call a matrix A ∈ Mn (C) positive semidefinite, and write A ≥ 0, if hAx, xi ≥ 0 for all x ∈ C2 . Let u, v ∈ Cn have norms less than 1, i.e., ||u|| < 1, ||v|| < 1. Note that |hu, vi| < 1. Show ! 1 1−hu,ui 1 1−hv,ui

1 1−hu,vi 1 1−hv,vi

≥ 0.

Exercise 1.0.22. (Zhang, pg. 112, #5.23)(6.2) Find all 2 × 2 complex matrices that are orthogonal, in the sense of hA, Bi = tr(B ∗ A) = 0, where A, B ∈ M2 (C), to the matrix 1 0 0 1 (a) (b) . 0 −1 1 0 Exercise 1.0.23. (6.2)(Petersen, pg. 166, #47) Let V be an inner product space and x1 , . . . , xn , y1 , . . . , yn ∈ V . Show the following generalized Cauchy-Schwarz inequality !2 ! n ! n n X X X |hxi , yi i| ≤ ||xi ||2 ||yi ||2 . i=1

i=1

i=1

Exercise 1.0.24. (6.3)(Tao, pg. 49, # 1.4.5) (Gram-Schmidt theorem) Let e1 , . . . , en be a finite orthonormal collection of vectors in a complex inner product space, and let v be a vector not in the span of e1 , . . . , en . Show that there exists a vector en+1 with span(e1 , . . . , en , en+1 ) = span(e1 , . . . , en , v) and e1 , . . . , en+1 is an orthonormal system.

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Hardest Problems Exercise 1.0.25. (6.1) The space of complex-valued continuous functions C([0, 1]) becomes an inner product space with the inner product Z 1 f (x)g(x) dx. hf, gi := 0

Justify that this does not define an inner product on the space of integrable functions L1 (0, 1]. Here, L1 [0, 1] is defined to be the vector space of sufficiently “nice” functions f : (0, 1] → C such that Z 1

|f (x)| dx < ∞. 0

For example, continuous functions are sufficiently “nice”. Exercise 1.0.26. (Zhang, pg. 110, #5.18)(6.1) Let V be a real inner product space. (a) If v1 , v2 , v3 , v4 ∈ V are pairwise product negative; that is, hvi , vj i < 0,

i, j = 1, 2, 3, 4, i 6= j,

show that v1 , v2 , v3 are linearly independent. (b) Is it possible for four vectors in the xy-plane to have pairwise negative products? How about three vectors? (c) Are v1 , v2 , v3 , v4 in part (a) necessarily linearly dependent or independent? (d) Suppose that u, v, and w are three unit vectors in xy-plane. What are the maximum and minimum values that hu, vi + hv, wi + hw, ui can attain? and when? Exercise 1.0.27. (6.2)(Petersen, pg. 166, #8) Let Rn be equipped with the standard inner product. Let S n−1 = {x ∈ Rn : ||x|| = 1} be the unit sphere. Given x, y ∈ S n−1 , we define the angle between x and y to be ](x, y) := arccos(x · y). When n = 1 it consists of 2 points. When n = 2, it is a circle etc. A finite subset {x1 , . . . , xk } ⊂ S n−1 is said to consist of equidistant points if ](xi , xj ) = θ for all i 6= j. (a) Show that this is equivalent to assuming that hxi , xj i = cos θ for all i 6= j. (b) Show that S 0 contains a set of two equidistinct points, S 1 a set of 3 equidistant points, and S 2 a set of four equidistant points. (c) Using induction on n show that a set of equidistant points in S n−1 contains no more than n + 1 elements. Exercise 1.0.28. (6.3)(Petersen, pg. 176, #11) (P. Enflo) Let V be a finite-dimensional inner product space and x1 , . . . , xn , y1 , . . . , yn ∈ V . Show Enflo’s inequality ! ! !2 n n n X X X |hxi , xj i|2 |hyi , yj i|2 . |hxi , yj i|2 ≤ i,j=1

i,j=1

i,j=1

Exercise 1.0.29. (6.2) (Zhang, pg. 60, #3.53) Let u, v ∈ Rn be nonzero column vectors orthogonal to each other, that is, v t u = 0. Find all eigenvalues of A = uv t and corresponding eigenvectors. Find also A2 . Is A similar to a diagonal matrix?


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Exercise 1.0.30. (6.1)(Tao) Show that h·, ·i is an inner product for Cn if and only if there exists an n × n positive semidefinite matrix A such that for all x, y ∈ Cn hx, yi = y ∗ Ax. Recall that we say a matrix A ∈ Mn (C) is positive semidefinite if hAx, xi ≥ 0 for all x ∈ Cn . Exercise 1.0.31. (6.3)(Petersen, pg. 175, #9) (Legendre Polynomials) Fix a < b in R. Consider the inner product Z b f (t)g(t) dt hf, gi = a

on R[t] and define q2n (t) = (t − a)n (t − b)n , dn pn (t) = n (q2n (t)). dt (a) Show that q2n (a) = q2n (b) = 0 .. . dn−1 dn−1 (a) = (b) = 0. dtn−1 dtn−1 (b) Show that pn has degree n. (c) Use induction on n to show that pn (t) is perpendicular to 1, t, . . . , tn−1 . Hint: Use integration by parts. (d) Show that p0 , p1 , . . . , pn , . . . are orthogonal to each other. References [1] Peter Petersen. Linear algebra. Los Angeles, CA, 2000. http://www.calpoly.edu/~ jborzell/Courses/Year%2010-11/Fall%202010/ Petersen-Linear Algebra-Math 306.pdf. [2] Terence Tao. An Epsilon of Room, I: Real Analysis. American Mathematical Society, Providence, RI, 2010. [3] Fuzhen Zhang. Linear algebra: challenging problems for students. 2nd ed. Johns Hopkins University Press, Baltimore, MD, 2009.