SWILA PROBLEM SET #10 SECTIONS 6.4-6.5 THURSDAY, JULY 14

Goals for Problem Set: • Use the background from Sections 6.1-6.3 to learn about orthogonal complements, orthogonal projections, and adjoints. • Given a subspace W ⊆ V , work with the orthogonal complement W ⊥ . • Apply the defining property of the adjoint: hT x, yiW = hx, T yiV . 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 4 “Easiest” problems. I like the first 4 “Mediumest” problems. I think the first 3 “Hardest” problems look interesting.

Easiest Problems Exercise 1.0.1. (Zhang, pg. 112, #5.22)(6.4) Consider R2 with the standard inner product. Let u = (1, 0) and v = (1, −1). Are they orthogonal? Find (a) (b) (c) (d) (e)

u⊥ , v ⊥ . u⊥ ∩ v ⊥ . {u, v}⊥ . (span{u, v})⊥ . span{u⊥ , v ⊥ }.

Exercise 1.0.2. (Zhang, pg. 82, #4.27)(6.5) Let A be an m×n matrix and x be an n-column vector. Show that (A∗ A)x = 0 ⇔ Ax = 0 Hint: Recall Rm , Rn becomes inner product spaces when endowed with the dot product, and consider hA∗ Ax, xi. 1

2

SECTIONS 6.4-6.5 THURSDAY, JULY 14

Exercise 1.0.3. (6.5) Given a linear transformation T : V → W , we proved that there is a map T ∗ : W → V satisfying hT x, yiW = hx, T ∗ yiV

for all x ∈ V, y ∈ W.

Complete the proof of the existence of the linear transformation T ∗ : W → V by showing T ∗ is linear. Hint: For α ∈ F, v ∈ V , and w1 , w2 ∈ W , analyze hv, αT ∗ w1 + T ∗ w2 i − α ¯ hv, T ∗ w1 i − hv, T ∗ w2 i by using the definition of the inner product. Then choose v appropriately. Exercise 1.0.4. (6.5) Recall Σc (N) is the vector space of real-valued sequences with only finitely many nonzero terms. Let S = {e1 , e2 , e3 , . . .} be the standard basis for Σc (N). Define the shift map S : Σc (N) → Σc (N) induced by S(ei ) = ei+1 , i.e., S(a1 , a2 , a3 , . . .) = (0, a1 , a2 , a3 , . . .) for all (a1 , a2 , a3 , . . .) ∈ Σc (N). (a) Confirm that S : Σc (N) → Σc (N) is a linear operator. (b) Show that Σc (N) becomes a real inner product space when endowed with the inner product ∞ X h(a1 , a2 , . . .), (b1 , b2 , . . .)i := aj b j . j=1

(Why is this sum finite?) (c) Prove that S is unitary in the sense that ||S(x)|| = ||x|| for all x ∈ Σc (N). (d) Find the adjoint S ∗ of S. In other words, construct the linear operator S ∗ : Σc (N) → Σc (N) such that hS(a1 , a2 , . . .), (b1 , b2 , . . .)i = h(a1 , a2 , . . .), S ∗ (b1 , b2 , . . .)i for all (a1 , a2 , . . .), (b1 , b2 , . . .) ∈ Σc (N). Hint: Consider a shift in the opposite direction. Exercise 1.0.5. (Zhang, pg. 108, #5.6)(6.4) Let  V = x = (x1 , x2 , x3 , x4 ) ∈ R4 : x1 = x3 + x4 , x2 = x3 − x4 . Show that V is a subspace of R4 . Find a basis for V and for V ⊥ . Exercise 1.0.6. (Zhang, pg. 114, #5.29)(6.4) If S is a subset of an inner product space V , answer true or false: (a) S ∩ S ⊥ = {0}. (b) S ⊆ (S ⊥ )⊥ . (c) (S ⊥ )⊥ = S. (d) (S ⊥ )⊥ = span(S). (e) S ⊥ = (span(S))⊥ .

SWILA PROBLEM SET #10

3

Exercise 1.0.7. (Zhang, pg. 116, #5.38)(6.5) If S and T are linear transformations from an inner product space V to an inner product space W such that hS(v), wi = hT (v), wi,

for all v ∈ V and w ∈ W,

show that S = T . Exercise 1.0.8. (Zhang, pg. 114, #5.26)(6.4) Let u ∈ V . Show v ∈ W is a projection of u onto W ; that is, u = v + v 0 , for some v 0 ∈ W ⊥ , if and only if ||u − v|| ≤ ||u − w||, for every w ∈ W. Exercise 1.0.9. (6.5) Prove the following corollary of the Fredholm Alternative: Let T : V → W be a linear map between finite-dimensional inner product spaces. If T is surjective, then T ∗ is injective. If T is injective, then T ∗ is surjective. In particular, if T is an isomorphism, then T ∗ is an isomorphism. Mediumest Problems Exercise 1.0.10. (6.4)(Zhang) Let V be an inner product space and W a nontrivial subspace of V . Let projW be the orthogonal projection onto W . (a) Find a nonidentity linear transformation P such that projW (w) = P (w),

w ∈ W,

but projW 6= P.

(b) Show that for every v ∈ V , hprojW (v), vi ≥ 0. (c) Show that I − projM is the orthogonal projection onto W ⊥ . Exercise 1.0.11. (Zhang, pg. 115, #5.30)(6.4) Let W1 and W2 be subspaces of an inner product space V . Show that (a) (W1 + W2 )⊥ = W1⊥ ∩ W2⊥ . (b) (W1 ∩ W2 )⊥ = W1⊥ + W2⊥ . Exercise 1.0.12. (6.5) Let B = {e1 , . . . , en } be an orthonormal basis for an inner product space V . Define the linear transformation   a1   ( e1 · · · en ) : Fn → V by ( e1 · · · en )  ...  = a1 e1 + · · · + an en . an Prove that the adjoint of ( e1 · · · en ) is given by ( e1 · · · en ) (x) = [x]B . Exercise 1.0.13. (6.5) (Zhang, pg. 116, #5.39) Use the Fredholm Alternative (from the notes) to prove the following: If T is a linear operator on V , show that V = ker T ∗ ⊕ Im(T ) = Im(T ∗ ) ⊕ ker T. Exercise 1.0.14. (Zhang, pg. 114, #5.29)(6.4) If S is a subset of an inner product space V , answer true or false: (a) [(S ⊥ )⊥ ]⊥ = S ⊥ .

4

SECTIONS 6.4-6.5 THURSDAY, JULY 14

(b) S ⊥ is always a subspace of V . (c) S ⊥ ⊕ span(S) = V . (d) dim(S ⊥ ) + dim(S ⊥ )⊥ = dim(V ). Exercise  1.0.15. (6.5) (Zhang, pg. 31, # 2.29) Let m and n be positive integers and denote Im 0 K = . Let SK be the collection of all (m + n)−square complex matrices X 0 −In such that X ∗ KX = K. ¯ A∗ ∈ SK . (1) If A ∈ SK , show that A−1 exists and A−1 , At , A, (2) If A, B ∈ SK , show that AB ∈ SK . How  about kA  or A + B? 0 Im (3) Discuss a similar problem with K = . −Im 0 Exercise 1.0.16. (6.4) Let V be an inner product space. Fix a subspace M ⊆ V and an orthonormal basis {v1 , . . . , vm } for M . We defined the orthogonal projection onto M by projM : V → V,

x 7→ hx, v1 iv1 + · · · + hx, vm ivm .

Show that this defintion of the orthogonal projection is independent of the choice of the 0 orthonormal basis for M ; i.e., if {v10 , . . . , vm } is another orthonormal basis for V , 0 0 projM (x) = hx, v10 iv10 + · · · + hx, vm ivm .

Hardest Problems Exercise 1.0.17. (Zhang, pg. 108, #5.8)(6.4) Define an inner product on P4 (R) over R as follows: Z 1 f (x)g(x) dx. hf, gi = 0

Let W be the subspace of P4 (R) consisting of the zero polynomial and all polynomials with degree 0; that is W = R. Find a basis for W ⊥ . What is the dimension of W ⊥ ? Exercise 1.0.18. (Zhang, pg. 112, #5.21)(6.4) Show that W = {X ∈ Mn (C) : tr(X) = 0} is a subspace of Mn (C) and W ⊥ consists of scalar matrices; that is, if tr(AX) = 0 for all X ∈ Mn (C) with tr(X) = 0, then A = λI for some scalar λ. Find the dimensions of W and W ⊥. Exercise 1.0.19. (6.4) Let E1 , E2 , . . . , Em be projections on an n-dimensional vector space V ; that is, Ei2 = Ei . (a) Show that if E1 + E2 + · · · + Em = I, then V = Im(E1 ) ⊕ Im(E2 ) ⊕ · · · Im(Em ) and Ei Ej = 0, i, j = 1, 2, . . . , m, i 6= j. (b) Given a subspace S ⊆ V , we call a linear operator E : V → V an orthogonal projection onto S if E(w) = w, w ∈ W and E(w0 ) = 0, w0 ∈ W ⊥ . Define an inner product for V such that each Ei is an othogonal projection.

SWILA PROBLEM SET #10

5

(c) Show that if Ei Ej = 0, i, j = 1, 2, . . . , m, i 6= j, then V = Im(E1 ) ⊕ Im(E2 ) ⊕ · · · ⊕ Im(Em ) ⊕

m \

! ker(Ei ) .

i=1

Exercise 1.0.20. (Zhang, pg. 63, #3.68)(6.5) Let A be a square complex matrix and denote ρ = max{|λ| : λ is an eigenvalue of A}, ω = max{|x∗ Ax| : x∗ x = 1, x ∈ Cn }, σ = max{(x∗ A∗ Ax)1/2 : x∗ x = 1, x ∈ Cn }. Show that ρ ≤ ω ≤ σ. Above by x ∈ C , x should be viewed as a column vector. n

Exercise 1.0.21. (Zhang, pg. 99, #4.111)(6.4) Given A ∈ Mn (C), we say A ≥ 0 if hAx, xi ≥ 0 for all x ∈ Cn . Given A, B ∈ Mn (C), we say B ≤ A if A−B ≥ 0. Finally, a square complex matrix X is called an orthogonal projection if X ∗ = X and X 2 = X. Let A and B be n × n matrices that are orthogonal projections. Show that B ≤ A if and only if AB = B. Exercise 1.0.22. (Zhang, pg. 80, #4.17)(6.5) Let A = (aij ) be an n × n real symmetric matrix, i.e., aij = aji for all i, j. Denote the sum of all entries of A by S(A). Show that S(A)/n ≤ S(A2 )/S(A). References [1] Fuzhen Zhang. Linear algebra: challenging problems for students. 2nd ed. Johns Hopkins University Press, Baltimore, MD, 2009.