SWILA PROBLEM SET #2 SECTIONS 2.3-2.6 THURSDAY, JUNE 9

Goals for Problem Set: • Figure out how coordinates are related to matrix representations • Relate matrices with linear transformations • Know the definitions of dimension and isomorphism 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. For some problems, I write “Good for all”. This means that whichever section the problem is located in, it might be useful for everyone to do it. 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. Finally, be aware that I don’t expect for you (or necessarily want you) to be able to complete all of the problems.

Make sure to introduce yourself to your neighbors!

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SECTIONS 2.3-2.6 THURSDAY, JUNE 9

Easiest Problems Exercise 2.0.1. (2.3) Let V = P3 (R) be the vector space of polynomials of degree at most 3 with real coefficients. Let S := {1, x, x2 , x3 } be the standard basis for V . Define the differentiation linear operator D : P3 (R) → P3 (R) by D(a0 + a1 x + a2 x2 + a3 x3 ) := a1 + 2a2 x + 3a3 x2 . Find the matrix representation [D]B . Exercise 2.0.2. (2.6) A finite vector space is a vector space with finitely many elements. Suppose F = Z/pZ for some positive prime p. (You just need to know that Z/pZ is a field with p elements.) Prove by counting that dimension is well-defined for finite vector spaces over F. In particular, show that that if a finite vector space V has a basis with n elements, then any other basis has n elements. Exercise 2.0.3. (2.3) (Zhang, pg. 58, #3.39)(Good for all) Find a 3 × 3 real matrix A such that Au1 = u2 , Au2 = 2u2 , Au3 = 3u3 , where       1 2 −2 u1 =  2  , u2 =  −2  , u3 =  −1  . 2 1 2 Hint: Does there exist a basis B for R3 for which [A]B is diagonal? Remarks: You can leave your answer as a product of matrices. You can also assume that the matrix   1 2 −2  2 −2 −1  2 2 2 is invertible. (I checked it.) Exercise 2.0.4. (2.3)(Petersen, pg. 33, #10) If T : V → V has a lower triangular representation with respect to the basis x1 , . . . , xn , show that it has an upper triangular representation with respect to xn , . . . , x1 . Exercise 2.0.5. (2.6)(Petersen, pg. 35, #2) Let T : V → W be a linear map. Show that T is an isomorphism if and only if it maps a basis for V bijectively to a basis for W . Exercise 2.0.6. (2.3)(Petersen, pg. 33, #9) Let e1 , e2 be the standard basis for C2 and consider the two real bases e1 , e2 , ie1 , ie2 and e1 , ie1 , e2 , ie2 . If λ = x + iy is a complex number, then compute the real matrix representation for λ1C2 with respect to both bases. Exercise 2.0.7. (2.6) Prove the following lemma from the notes: A linear transformation T : V → W is an isomorphism if and only if it is bijective.

SWILA PROBLEM SET #2

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Mediumest Problems Exercise 2.0.8. (2.3 & 2.6) Recall Mn (F) is defined to be the vector space of n × n matrices with entries in F. Suppose T : M3 (F) → M4 (F) is a linear transformation. Let B be a basis of M3 (F) and C a basis of M4 (F). What is the size of the matrix [T ]C,B ? For example, is it 3 × 4, 4 × 9, or something else? Exercise 2.0.9. (2.3) Let V be a vector space with an ordered basis B = {v1 , v2 , . . . , vn }. Define the ordered basis B 0 = {vn , vn−1 , . . . , v1 }. Given a linear operator T : V → V , how do the matrices [T ]B and [T ]B0 compare? In other words, if we write   a11 a12 · · · a1n  a21 a22 · · · a2n   [T ]B =   ··· ··· ··· ··· , an1 an2 · · · ann what is [T ]B0 ? Exercise 2.0.10. (Zhang, pg. 30, # 2.19)(2.4) Find the inverse of the matrices   1 1 1  0 1 1  and 0 0 1



0  0   0 1

0 0 1 0

0 1 0 0

 1 0  . 0  0

Hint: Think of these matrices as change of coordinate matrices. See the example in Section 2.4. Exercise 2.0.11. (Zhang, pg. 11, # 1.12)(2.6) Recall that P2 (R) is defined to be the space of polynomials with real coefficients of degree at most 2. Show that {1, x − 1, (x − 1)(x − 2)} is a basis of P2 (R) and that W = {p(x) ∈ P2 (R) : p(1) = 0} is a subspace of P2 (R), i.e., W is closed under vector addition and scalar multiplication. Find dim(W ). Exercise 2.0.12. (2.3) Let A = (aij ) ∈ Mn (F). Show that (a)         a1n a12 a11 x1  a2n   a21   a22   x2          A  .  = x1  .  + x2  .  + · · · + xn  .  . . . . .  .   .   .   .  ann an2 an1 xn n (b) For all v1 , v2 , . . . , vm ∈ F ,     | | | | | | A  v1 v2 · · · vm  =  A(v1 ) A(v2 ) · · · A(vm )  . | | | | | | Exercise 2.0.13. (2.3) (Lang, pg. 97, # 18) Let T : R2 → R2 be a linear map such that T (4, 1) = (1, 1) and T (1, 1) = (3, −2). If S is the standard basis for R2 , compute [T ]S .

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SECTIONS 2.3-2.6 THURSDAY, JUNE 9

Exercise 2.0.14. (2.4)(Zhang, pg. 30) Let inverse of the matrix  0 a1 0  0 0 a2   .. .. . .  . . .   0 0 0 an 0 0 Hint: Think of it as a change of a coordinate

a1 , a2 , . . . , an be nonzero numbers. Find the  ··· 0 ··· 0   ..  . .. . .   · · · an−1  ··· 0 matrix. See the example in Section 2.4.

Exercise 2.0.15. (Zhang, pg. 19, # 1.44)(2.6) Let    z   1      z 2  4  ∈ C : z = z + z and z = z − z . W =  3 1 2 4 1 2 z3        z 4

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(a) View C as a complex vector space. Prove that W is a subspace of C4 , i.e., W is closed under vector addition and scalar multiplication. (b) Find a basis for W . What is the dimension of W . (c) Prove that S := {k(1, 0, 1, 1) : k ∈ C} is a subspace of W , i.e., S ⊂ W is closed under addition and scalar multiplication. Exercise 2.0.16. (Zhang, pg. 56, #3.29)(2.5) Let A, B ∈ Mn (C). If AB = 0, show that for any positive integer k, tr((A + B)k ) = tr(Ak ) + tr(B k ). Exercise 2.0.17. (Zhang, pg. 60, #3.55)(2.5) Let A, B ∈ Mn (C). Show that tr((AB)k ) = tr((BA)k ). Is it true in general that tr((AB)k ) = tr(Ak B k )? Exercise 2.0.18. (Zhang, pg. 35, # 2.43)(2.6) Let A ∈ Mn (C) and u1 , u2 , . . . , un ∈ Cn be linearly independent. Show that A is invertible if and only if Au1 , Au2 , . . . , Aun are linearly independent. Exercise 2.0.19. (2.6)(Petersen, pg. 36, #5) Fix a vector space V 6= {0} and linear operators T1 , . . . , Tn on V . Show that if T1 ◦ · · · ◦ Tn = 0, then Ti is not one-to-one for some i = 1, . . . , n. Is it necessarily true that Ti = 0 for some i?

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Hardest Problems Exercise 2.0.20. (2.4)(Suggested by Alyssa Loving) In this exercise, we will explore change of coordinate matrices. Suppose B = {v1 , v2 } and C = {w1 , w2 } are two bases for R2 . Show that the matrix that changes coordinates with respect to B to coordinates with respect to C is given by  −1   | | | | A =  w1 w2   v1 v2  ; | | | | more precisely, 

   x x =A for all (x, y) ∈ R2 . y y C B More generally, suppose B 0 = {v1 , v2 , . . . , vn } and C 0 = {w1 , w2 , . . . , wn } are bases for Fn . Show that the matrix  −1   | | | | | | B =  w1 w2 · · · wn   v1 vw · · · vn  | | | | | | changes coordinates with respect to B 0 to coordinates with respect to C 0 . Hint: Draw a commutative diagram. Exercise 2.0.21. (2.6)(Petersen, pg. 36, #6) Recall that Pn (C) is the complex vector space of polynomials of degree at most n. Let t0 , . . . , tn ∈ R be distinct. Define T : Pn (C) → Cn+1 by T (p) = (p(t0 ), . . . , p(tn )). Show that T is an isomorphism. (This problem will be easier to solve after we prove the Rank-Nullity Theorem.) Exercise 2.0.22. (2.6)(Petersen, pg. 36, #13) Show that Mn (C) is a real vector space of dimension n2 and M2n (R) is a real vector space of dimension 4n2 . Conclude that there must be matrices in M2n (R) that do not come from complex matrices in Mn (C). Find an example of a matrix in M2 (R) that does not come from M1 (C). Exercise 2.0.23. (2.6)(Petersen, pg. 36, #7) Let t0 ∈ F and consider Pn (C) ⊂ C[t]. Show that T : Pn (C) → Cn+1 defined by T (p) = (p(t0 ), p0 (t0 ), . . . , p(n) (t0 )) is an isomorphism. Hint: Think of a Taylor expansion at t0 . Exercise 2.0.24. (Zhang, pg. 9, # 1.3)(2.6) Define the vector space C2 = C × C = {(x, y) : x, y ∈ C}. Is C2 a vector space over C? Over R? Over Q? If so, find the dimension of C2 . Exercise 2.0.25. (2.6) (Zhang, pg. 65, #3.79) Let 0 6= A ∈ Mn (C). Define a linear operator T on Mn (C) by T (X) = AX − XA, X ∈ Mn (C). Confirm that T is in fact linear. Show that T is not invertible. We will later see that this is equivalent to 0 being an eigenvalue of T . Exercise 2.0.26. (Zhang, pg. 56, #3.28)(2.5) Let A ∈ Mn (C). Show that An = 0 if tr(Ak ) = 0, k = 1, 2, . . . , n.

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SECTIONS 2.3-2.6 THURSDAY, JUNE 9

Exercise 2.0.27. (Zhang, pg. 31, # 2.24)(2.6) Find the inverse of the 3 × 3 Vandermonde matrix   1 1 1 V =  a1 a2 a3  , a21 a22 a23 when a1 , a2 , and a3 are distinct from each other. 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.