SWILA PROBLEM SET #14 SECTIONS 8.1-8.2 THURSDAY, JULY 28

Goals for Problem Set: • Prove some functions are linear functionals. • Use that a certain collection of simple tensors form a basis for the tensor product. • Construct linear transformations using the tensor product structure. • Prove maps invovling tensor products are isomorphisms. 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 3 Easiest problems. I like the first 5 Mediumest problems. I like the first 3 Hardest problems.

Easiest Problems Exercise 1.0.1. (Zhang, pg. 118, #5.47)(8.1) Let V be an inner product space over F. (a) For v ∈ V , define a mapping Tv from V to F by Tv (u) = hu, vi,

for all u ∈ V.

Show that Tv ∈ V ∗ . (b) Let T be the mapping from V to V ∗ defined by for all v ∈ V.

T (v) = Tv ,

Show that T is linear. (c) Show that T is one-to-one, and hence, is an isomorphism. Exercise 1.0.2. (Lang, pg. 308, #1)(8.2) Let V , W be finite-dimensional spaces over F. Let F : V ⊗ W → U be a linear map. Show that the map (v, w) 7→ F (v ⊗ w) is a bilinear map of V × W into U . 1

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SECTIONS 8.1-8.2 THURSDAY, JULY 28

Exercise 1.0.3. (8.1) (a) Using the sequence definition of continuity, prove that every linear functional λ ∈ (Rn )∗ is continuous (using the standard metric on Rn ). (b) Recall that the norm of a linear transformation T : V → F is given by: ||T || = sup |T (x)|. ||x||=1 n ∗

Prove that (R ) = R in the following sense: For all f ∈ (Rn )∗ , there exists y = (y1 , . . . , yn ) ∈ Rn such that ||f ||V ∗ = |y| and n

f (x1 , . . . , xn ) = x1 y1 + · · · + xn yn

for all (x1 , . . . , xn ) ∈ Rn .

Hint: Let yj = f (ej ). Exercise 1.0.4. (Lang, pg. 165, #5)(8.1) Let V be a vector space of finite dimension n over the field F. Let ϕ be a linear functional on V , and assume ϕ 6= 0. What is the dimension of the the kernel of ϕ? Justify your answer. Exercise 1.0.5. (Lang, pg. 165, #6)(8.1) Let V be a vector space of dimension n over the field F. Let f, g be two non-zero linear functionals on V . Assume that there is no element λ ∈ K, λ 6= 0 such that f = λg. Show that (ker f ) ∩ (ker g) has dimension n − 2. Exercise 1.0.6. (8.1) Prove the following proposition from the notes: Let T, T˜ : V → W and S : W → U be linear transformations. Then (1) (aT + bT˜)∗ = aT ∗ + bT˜∗ (2) (S ◦ T )∗ = T ∗ ◦ S ∗ . Note that ∗ denotes a dual map here, not an adjoint. Exercise 1.0.7. (8.1)(Petersen, pg. 80, #1) Let x1 , . . . , xn be a basis for V and f1 , . . . , fn a dual basis for V ∗ . Show that the inverses to the isomorphisms [ x1 · · · xn ] : F n → V [ f 1 · · · f n ] : Fn → V ∗ are given by 

 f1 (x)   [ x1 · · · xn ]−1 (x) =  ...  fn (x)   f (x1 )   [ f1 · · · fn ]−1 (f ) =  ...  . f (xn ) Exercise 1.0.8. (8.1) Prove the following proposition from the notes: Fix a finite-dimensional {v1 , . . . , vn } be a basis for V . For each i, Pn vector space V . Let ∗ ∗ define vi : V → F by j=1 aj vj 7→ ai . Then {v1 , . . . , vn∗ } forms a basis for V ∗ . In particular, V ∗ is finite-dimensional with dim(V ∗ ) = dim(V ).

SWILA PROBLEM SET #14

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Mediumest Problems Exercise 1.0.9. (Lang, pg. 311, #1)(8.2) Let V, W be finite-dimensional vector spaces over F. Show that there is a unique isomorphism of V ⊗ W on W ⊗ V sending v ⊗ w to w ⊗ v for all v ∈ V and w ∈ W . Hint: For uniqueness, note a certain subset of the simple tensors form a basis for the tensor product. Exercise 1.0.10. (Lang, pg. 308, #2)(8.2) Show that the correspondence g 7→ g∗ coming from the Universal Property of Tensor Products is an isomorphism of bilinear maps of V ×W into U , and the space of linear maps L(V ⊗ W, U ). Exercise 1.0.11. (Lang, pg. 308, #3)(8.2) Let V be a finite-dimensional vector space over F. Let T : V → V be a linear map. Show that there exists a unique linear map F : V ⊗ V → V ⊗ V such that F (v ⊗ w) = T v ⊗ T w for all v, w ∈ V . This map is denoted by T ⊗ T . Exercise 1.0.12. (Lang, pg. 308, #4)(8.2) Generalize the previous exercise to a tensor product V ⊗ W . Let S : V → V and T : W → W be linear maps. Show how to define the linear map S⊗T :V ⊗W →V ⊗W satisfying S ⊗ T (v ⊗ w) = Sv ⊗ T w for all v ∈ V , w ∈ W . Exercise 1.0.13. (Lang, pg. 311, #2)(8.2) Let V, W be finite-dimensional vector spaces over F. Show that there is a unique isomorphism V ∗ ⊗ W → L(V, W ) such that f ⊗ w 7→ Tf,w , where Lf,w is the unique map such that Tf,w (v) = f (v)w. Exercise 1.0.14. (Lang, pg. 165, #3)(8.1) Let V be a finite-dimensional vector space over the field F. Let U , W be subspaces, and assume that V is the direct sum U ⊕ W . Show that V ∗ is equal to the direct sum U ∗ ⊕ W ∗ . (In the book, it was written U ⊥ ⊕ W ⊥ , but I think it’s supposed to be this.) Exercise 1.0.15. (8.1)(Petersen, pg. 81, #4) Let f, g ∈ V ∗ and assume that g 6= 0. Show that f = λg for some λ ∈ F if and only if ker(f ) ⊇ ker(g). Exercise 1.0.16. (8.1)(Petersen, pg. 81, #10) (The Rank Theorem) Let T : V → W and x1 , . . . , xk a basis for Im(T ). (a) Show that T (x) = f1 (x)x1 + · · · + fk (x)xk for suitable f1 , . . . , fk ∈ V ∗ .

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SECTIONS 8.1-8.2 THURSDAY, JULY 28

(b) Show that T ∗ (f ) = f (x1 )f1 + · · · + f (xk )fk for f ∈ W ∗ . Here, T ∗ is the dual map of T . (c) Conclude that rank(T ∗ ) ≤ rank(T ). (d) Show that rank(T ∗ ) = rank(T ). Exercise 1.0.17. (8.1) Definition. A normed space V is a (C or R) vector space with a norm || · || : V → R ≥ 0 satisfying: For all x, y ∈ V and λ ∈ C, • (Non-degeneracy) ||x|| > 0 if x 6= 0. • (Homogeneity) ||λx|| = |λ| · ||x||. • (Triangle inequality) ||x + y|| ≤ ||x|| + ||y||. Let V be a normed space (not necessarily finite-dimensional). Fix a linear functional f ∈ V ∗ . Prove the following are equivalent: (1) f is continuous at all x ∈ V , i.e., limy→x f (y) = f (x) for all x ∈ V . (2) f is continuous at 0. |f (x)| (3) sup < ∞. x6=0 ||x|| We say that f is a continuous linear functional if any of the above three (equivalent) conditions hold and define |f (x)| ||f ||V ∗ := sup . x6=0 ||x|| |f (x)| = sup |f (x)|.) (Note that sup x6=0 ||x|| ||x||=1 √ Hint: If (3) does not hold, show that there is a sequence (x ) in V such that ||x || = 1/ n i i √ and |f (T xi )| ≥ n. Exercise 1.0.18. (8.1) Let V be a normed space. For v ∈ V , we define ι(v) ∈ V ∗∗ by ι(v)(f ) = f (v). A corollary of the Hahn-Banach Theorem states that for every v ∈ V , there exists f ∈ V ∗ such that f (v) = v and ||f ||V ∗ = 1. Prove that ||ι(v)||V ∗∗ = ||v||. Exercise 1.0.19. (8.1) Recall we defined Σc (N) to be the space of compactly supported real sequences. We define l1 (N) to be the vector space of absolutely summable real sequences. Show that Σc (N)∗ = l1 (N) in the sense that every continuous linear functional f ∈ Σc (N)∗ can be canonically identified with an element (xn ) ∈ l1 (N) such that ∞ X f ((yn )) = yn xn , for all (yn ) ∈ Σc (N), and n=1

||f ||Σc (N)∗ =

∞ X n=1

|xn |.

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Hardest Problems Exercise 1.0.20. (8.2) Recall that one can prove using Zorn’s Lemma that every vector space has a basis. Consider the construction of the tensor product of two finite-dimensional vector spaces, the proof that this tensor product satisfies a universal property, and that one can use bases of the individual vector spaces to find a basis for the tensor product. Sans the statement that dim(V ⊗ W ) = dim(V )dim(W ), consider whether one can reuse the finite-dimensional proof to prove the theorem for general vector spaces, i.e., not necessarily finite-dimensional. Exercise 1.0.21. (Lang, pg. 311, #3)(8.2) Let V, W be finite-dimensional vector spaces over F. Show that there is a unique isomorphism V ∗ ⊗ W ∗ → (V ⊗ W )∗ which to each tensor product f ⊗ g (f ∈ V ∗ and g ∈ W ∗ ) associates a linear functional Tf,g of V ⊗ W having the property that Tf,g (v ⊗ w) = f (v)g(w). Describe this isomorphism in terms of bases and dual bases. Exercise 1.0.22. (8.1)(Petersen, pg. 81, #6) Let V and W be finite-dimensional vector spaces. Exhibit an isomorphism between V ∗ × W ∗ and (V × W )∗ that does not depend on choosing bases for V and W . Exercise 1.0.23. (Petersen, pg. 81, #3)(8.1) Given the basis 1, t, t2 for P2 (C), identify P2 (C) with C3 and (P2 )∗ with M1×3 (C). (a) Using these identifications, find a dual basis to 1, 1 + t, 1 + t + t2 in (P2 )∗ . (b) Fix t0 ∈ C. Using these identifications, find the matrix representation for f ∈ (P2 )∗ defined by f (p) = p(t0 ). (c) Fix a < b in R. Using these identifications, find the matrix representation for f ∈ (P2 )∗ defined by Z b p(t) dt. f (p) = a ∗

(d) Are all elements of (P2 ) represented by the types described in either (b) or (c) (up to a linear combination of the two)? Exercise 1.0.24. (8.1) Let T : V → W be a linear transformation of finite-dimensional inner product spaces. In what way (if any) can we identity the adjoint of T (defined in Chapter 6) with the dual map of T ? Exercise 1.0.25. (8.1) We may view V ⊆ V ∗∗ via an injective linear transformation ι : V → V ∗∗ . For v ∈ V , we define ι(v) ∈ V ∗∗ by ι(v)(f ) = f (v). If every element of the double dual V ∗∗ is of the form ι(v), v ∈ V , we say V is reflexive. Give an example of (infinite-dimensional) normed space that is not reflexive. However, show that every finite-dimensional normed space is reflexive.

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SECTIONS 8.1-8.2 THURSDAY, JULY 28

References [1] Serge Lang. Linear Algebra. 2nd ed. Addison-Wesley, Reading, MA, 1971. [2] 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. [3] Fuzhen Zhang. Linear algebra: challenging problems for students. 2nd ed. Johns Hopkins University Press, Baltimore, MD, 2009.