# Math 4326 Linear Algebra Homework

## Dr. Duval

### Complex NumbersDefinition of Vector Space

(pp. 1-10)
A: Reading questions. Hand in Thu. 19 Jan., or earlier.
1. Verify, using properties of real numbers, and that i^2=-1, that complex numbers satisfy the distributive property. [Note typo about equation for i in handout!!]
2. What does F stand for?
3. The picture for addition in F^n on p. 7 is 2-dimensional (n=2), since it is drawn on a 2-dimensional piece of paper. Does this picture work for larger values of n? Why or why not?
4. What gets multiplied in scalar multiplication?
5. Verify associativity in C^n.
6. Verify commutativity in F^infinity.
B: Warmup exercises. For you to present in class. Due in class Thu. 19 Jan.
• Verify distributivity in P(F).
• Ch. 1: 2.
C: Main exercises. Hand in at beginning of class, Thu. 26 Jan.
• Verify that F^infinity is a vector space.
• Verify that P(F) is a vector space. [Grad students only]

### Properties of Vector SpacesSubspaces

(pp. 11-14).
A: Reading questions. To hand in by Mon., 23 Jan.
1. In Proposition 1.3, why do we "[s]uppose that w and w' are additive inverses of v"?
2. In Propositions 1.4 and 1.5, identify which properties of vector spaces are used at each step of the proof.
3. What is the difference between Propositions 1.4 and 1.5?
4. In the middle of p. 13, the text claims that to check if a subset U of vector space V is a subspace, we only need to check 3 conditions, instead of all the conditions listed on p. 9. It then gives examples of how the 3 conditions on p. 13 imply some of the other conditions on p. 9. Pick two other conditions on p. 9, and show how the conditions on p. 13 lead to them.
5. Check that the two examples at the bottom of p. 13 are indeed subspaces, as claimed in the text.
B: Warmup exercises. For you to present in class. Due in class Tue. 24 Jan.
• Ch. 1: 3, 5, 6.
C: Main exercises. Hand in at beginning of class, Thu Feb. 2.
• Ch. 1: 8.

### Sums and Direct Sums

(pp. 14-18).
A: Reading questions. Hand in at beginning of class, Tue. 24 Jan.
1. Verify that U_1 + ... + U_m is a subspace.
2. Verify equation 1.7. [Note that the text uses 1.7 twice, each with a different choice of W. Pick one choice of W to verify equation 1.7 for.]
3. Does U + W exist for any pair of subspaces U and W? Does U "oplus" W exist for any pair of subspaces U and W? Justify your answer in each case. [Note that "oplus" is how I am transcribing the symbol that is a plus sign inside a circle, and which stands for direct sum.]
4. Verify F^3 = U "oplus" W in the example in the middle of p. 15.
5. Verify P(F) = U_e "oplus" U_o, as claimed at the top of p. 16.
6. In the proof of Proposition 1.8, where do we use the assumption that the U_i's are subspaces? [Note: This may be at just one point in the proof, or at more than one point.]
7. Which do you think will be more useful for establishing a direct sum equation, Proposition 1.8, or Proposition 1.9? Justify your answer. If you cannot decide, then present arguments in favor of both.
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 26 Jan.
• Ch. 1: 10, 11, 13.
C: Main exercises. Hand in at beginning of class, Thu Feb. 2.
• Ch. 1: 14.
• Ch. 1: 15. [Grad students only]

### Span and Linear Independence

(pp. 22-27).
A: Reading questions. Hand in at beginning of class, Thu. 26 Jan.
1. Verify that the span of any list of vectors in V is a subspace of V.
2. Verify that, if some vectors are removed from a linearly independent list, then the remaining list is also linearly independent.
3. Demonstrate Lemma 2.4 on the linearly dependent list from the middle of p. 24, (2,3,1), (1,-1,2), (7,3,8). In other words, find the v_j that makes (a) and (b) true, and show why (a) and (b) are in fact true in this case. [Hint: Use the proof.]
4. Demonstrate Lemma 2.6 on that same linearly dependent list from the middle of p. 24. [Note: This question was completely improperly phrased, and should be ignored.]
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 31 Jan.
• Verify that P_m(F) is a subspace of P(F)
• Ch. 2: 1, 2, 4.
C: Main exercises. Hand in at beginning of class, Thu Feb. 9.
• Ch. 2: 3.
• Ch. 2: 5. [Grad students only]

### Bases

(pp. 27-31).
A: Reading questions. Hand in at beginning of class, Tue. 31 Jan.
1. A note in the margin of p. 27 claims that the proof of Proposition 2.8 is "essentially a repetition of the ideas that led us to the definition of linear independence." Compare and contrast the proof of Proposition 2.8 to the the ideas leading to linear independence.
2. Verify that the process in the proof of Theorem 2.10 produces ((1,2),(4,7)) when applied to the list ((1,2),(3,6),4,7),(5,9)), as suggested on p. 29. Also verify that ((1,2),(4,7)) is indeed a basis of F^2.
3. Verify that the list ((2,3,1),(1,-1,2)) [the first two vectors from the list in the middle of p. 24] in F^3 can be extended to a basis in F^3, as promised by Theorem 2.12. [Hint: Use the proof of Theorem 2.12.]

Then, similarly, verify Proposition 2.13, using the span of ((2,3,1),(1,-1,2)) for U, and using F^3 for V.

B: Warmup exercises. For you to present in class. Due by the end of class Thu. 2 Feb.
• Ch. 2: 8, 9.
C: Main exercises. Hand in at beginning of class, Thu Feb. 9.
• Ch. 2: 10.

### Dimension

(pp. 31-34).
A: Reading questions. Hand in at beginning of class, Thu. 2 Feb.
1. What is the significance of Theorem 2.14? Why must it be the first result of this section? [Hint: What is the name of this section?]
2. Find the definition of "finite dimensional" vector space in the text. [Hint: Believe it or not, it is not in this section!] How does it compare to the definition of "dimension" in this section? Why are these two definitions compatible?
3. Which do you think will prove to be more useful, Proposition 2.16, or Proposition 2.17? Why?
4. The proof of Proposition 2.19 uses the assumption that equation 2.20 holds in order to show that a certain list of vectors spans V. [See the second sentence of the proof.] Verify that this list does span V.
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 7 Feb.
• Ch. 2: 11, 12, 13.
C: Main exercises. Hand in at beginning of class, Thu Feb. 16.
• Ch. 2: 16.
• Ch. 2: 17. [Grad students only]

### Definitions and Examples

(pp. 38-41).
A: Reading questions. Hand in at beginning of class, Tue. 7 Feb.
1. Verify the following functions, described on pp. 38-39, are in fact linear maps: identity, differentiation, multiplication by x^2, backward shift.
2. Near the top of p. 40, the text claims of a certain equation: "Because (v_1,...,v_n) is a basis of V, the equation above does indeed define a function T from V to W." Provide the missing details of this claim.
3. Verify that S+T is a linear map from V to W whenever S, T element-of L(V,W).
4. Verify the first distributive property on p. 41: (S_1 + S_2)T = S_1 T + S_2 T whenever T element-of L(U,V) and S_1, S_2 element-of L(V,W).
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 9 Feb.
• Ch. 3: 1, 2.
C: Main exercises. Hand in at beginning of class, Thu Feb. 16.
• Ch. 3: 3.

### Null Spaces and Ranges (part I)

(pp. 41-44).
A: Reading questions. Hand in at beginning of class, Thu. 9 Feb.
1. Find the null space of the identity map, defined on p. 38. Is this map injective? Why or why not?
2. Find the range of the backward shift map, defined on p. 39. Verify this map is surjective, as claimed on p. 44.
3. Does surjectivity of a map T element-of L(V,W) depend on V, W, both, or neither? If it does depend on V and/or W, give an example showing how changing V and/or W changes the surjectivity of T.
4. Now make a guess as to whether injectivity of a map T element-of L(V,W) depends on V and/or W. [Hint: This is not explicitly addressed in the book. Make your best guess, and provide some justification, not necessarily a proof, why you think this way.]
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 14 Feb.
• Ch. 3: 5, 6.
C: Main exercises. Hand in at beginning of class, Thu Feb. 23.
• Ch. 3: 7.

### Null Spaces and Ranges (part II)

(pp. 45-47).
A: Reading questions. Hand in at beginning of class, Tue. 14 Feb.
1. In the proof of Theorem 3.4, what are m and n, and how do we know dim V = m + n? How do we compute dim null T and dim range T? [Note: Theorem 3.4 is the most important theorem of the first four chapters of the book, and also has one of the longest proofs in these chapters. You can answer these reading questions just from carefully reading and understanding the first paragraph of the proof, which is all I ask you to do, though, of course, you are welcome to read the rest of the proof.]
2. In the proof of Corollary 3.5, there is a string of equalities and inequalities. The middle line of this string reads ">= dim V - dim W". Explain why ">=" is the correct relation here. [Note that ">=" is how I am transcribing the symbol for "greater than or equal".]
3. Near the top of p. 47, the text claims "we can rewrite the equation Tx=0 as a system of homogeneous equations". Why?
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 16 Feb.
• Ch. 3: 9, 12.
C: Main exercises. Hand in at beginning of class, Thu Feb. 23.
• Ch. 3: 13.
• Ch. 3: 11. [Grad students only]

### The Matrix of a Linear Map

(pp. 48-53).
A: Reading questions. Hand in at beginning of class, Thu. 16 Feb.
1. Pick bases of P_m(F) and P_(m-1)(F). [Hint: Pick nice bases!] Write out M(T), where T element-of L(P_m(F),P_(m-1)(F)) is the differentiation linear map, i.e., Tp= p'.
2. Verify equation 3.10.
3. In the long string of equalities in the middle of p. 51, what are u_k, v_r, and w_j?
4. Why define M(v) as it is defined in equation 3.13, as opposed to the more simple definition given for M(x) a little lower in the same page?
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 21 Feb.
• Ch. 3: 17, 18, 19.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 2.
• Ch. 3: 21.

### Invertibility

(pp. 53-58).
A: Reading questions. Hand in at beginning of class, Tue. 21 Feb.
1. When, if ever, in the proof of Proposition 3.17 do we use the linearity of T or of any other map?
2. In the second half of the proof of Theorem 3.18 (same dimension implies isomorphism), an invertible linear map T is defined. In this set-up, what is T(v_i)?

Recall that dim P_m(F) = m+1. Thus, Theorem 3.18 guarantees that P_m(F) is isomorphic to F^(m+1). Now pick nice bases of P_m(F) and F^(m+1), and describe the invertible linear map T that shows they are isomorphic.

3. Why is it necessary to explicitly mention the bases (v_1,...,v_n) and (w_1,...,w_m) for V and W, respectively, in the statement of Proposition 3.19? [Hint: Recall the definition of M.]
4. An easy corollary to Proposition 3.19 is that two vector spaces are isomorphic. Which ones, and why?
5. Theorem 3.21 claims that 3 statements, (a), (b), and (c), are equivalent. Yet, while the proof directly shows that (a) implies (b), it does not show directly that (b) implies (a). Why is this acceptable? How do we know (b) implies (a)?
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 23 Feb.
• Ch. 3: 20, 22.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 2.
• Ch. 3: 23.
• Ch. 3: 24. [Grad students only]

### Polynomials

(pp. 63-72). Note the summary on p. 63. Indeed, you are not responsible for any of the proofs in this chapter, but you should become familiar with the statements of all the results.
A: Reading questions. Hand in at beginning of class, Thu. 23 Feb.
1. Pick a polynomial of degree 3. Demonstrate Proposition 4.1 on your polynomial. That is, find a root lambda (be sure to demonstrate it's a root), and the corresponding polynomial q(x). [Hint: plan ahead!]
2. Pick an m >= 4 [m greater than or equal to 4]. Find a polynomial p with degree m such that p has less than m distinct roots.
3. Why does Corollary 4.8 have to include the phrase "(except for the order of the factors)"?
5. Describe as clearly as you can the differences between factorization in P(C) and factorization in P(R). [Hint: Focus on Corollary 4.8.]
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 28 Feb.
• Ch. 4: 1, 5.
C: Main exercises.
• none

### Invariant Subspaces

(pp. 76-79).
A: Reading questions. Hand in at beginning of class, Tue. 28 Feb.
1. Why are invariant subspaces important?
2. How is equation 5.3, which defines eigenvalues and eigenvectors, connected with one-dimensional invariant subspaces?
3. Does the choice of F affect the eigenvalues and eigenvectors of a linear transformation? If so, give an example; if not, explain why not.
4. Fill in the missing details of how we get from equation 5.8 to the next (unnumbered) displayed equation, lambda_k v_k = a_1 lambda_1 v_1 + ... + a_(k-1) lambda_(k-1) v_(k-1).
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 2 Mar.
• Ch. 5: 1, 5, 6.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 23.
• Ch. 5: 2.
• Ch. 5: 4. [Grad students only]

### Polynomials Applied to Operators Upper-Triangular Matrices (part I)

(pp. 80-83).
A: Reading questions. Hand in at beginning of class, Thu. 2 Mar.
1. Why doesn't T^m make sense when T is a linear map, but not a linear operator?
2. Theorem 5.10 is the most important result of Chapter 5, but its proof, while very clever, is not very complicated at all. Please try your best to understand it. Where in the proof do we use that the vector is complex (i.e., F=C)?
3. What does the proof of Theorem 5.10 have in common with the proof of Theorem 5.6? [Note: Compare the proofs, not the statements, of these theorems.]
4. Give three examples of a 3-by-3 upper-triangular matrix.
5. The author introduces upper-triangular matrices as nice ones to represent linear transformations. Given a fixed linear transformation (in other words, you don't get to pick the linear transformation), how can you represent it by an upper-triangular matrix? In other words, what can you pick cleverly to make sure the fixed linear transformation is represented by an upper-triangular matrix?
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 7 Mar.
• Ch. 5: 9, 10.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 23.
• Ch. 5: 8, 14.

### Upper-Triangular Matrices (part II)

(pp. 83-87).
A: Reading questions. Hand in at beginning of class, Tue. 7 Mar.
1. Demonstrate parts (b) and (c), for k=3, of Proposition 5.12 on the 4-by-4 upper triangular matrix near the top of p. 83. In other words, show that Tv_3 element-of span(v_1,...,v_3) and that span(v_1,...,v_3) is invariant under T. Note that the basis here is the standard basis.
2. In the middle of the second paragraph of the proof of Theorem 5.13, the text claims that, if lambda is an eigenvalue of T, then T - lambda I is not surjective by (3.21). Fill in the missing details of this claim.
3. Let T element-of L(V). How does finding a basis of V for which the matrix of T is upper triangular help find the eigenvalues of T? How does it help determine whether or not T is invertible? Demonstrate your answers on the 4-by-4 upper triangular matrix near the top of p. 83 (in other words, find the eigenvalues of that matrix, and also determine whether or not it is invertible).
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 21 Mar.
• Ch. 5: 18, 19.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 30.
• Ch. 5: 17.

### Diagonal Matrices

(pp. 87-90).
A: Reading questions. Hand in at beginning of class, Tue. 21 Mar.
1. Verify the claim at the bottom of p. 87 that an operator T element-of L(V) has a diagonal matrix (with lambda_1, ... , lambda_n on the diagonal and 0's elsewhere) with respect to a basis (v_1, ... ,v_n) of V if and only if
T v_1 = lambda_1 v_1; ... ; T v_n = lambda_n v_n.
2. Verify that the only eigenvalue of the linear operator T given in equation 5.19 is 0, and the the corresponding set of eigenvectors is only 1-dimensional. In what way is this surprising?
3. Verify that T element-of L(F^3) defined by T(z_1, z_2, z_3) = (4z_1,4z_2,5z_3), as on p. 88, satisfies each of the five conditions (a)-(e) of Proposition 5.21. You may need to read the proof to help you with some of these, though I don't think you need to fully understand the proof in order to complete the verification of (a)-(e).
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 23 Mar.
• Ch. 5: 11.
C: Main exercises. Hand in at beginning of class, Thu. Mar. 30.
• Ch. 5: 20
• Ch. 5: 21.

### Inner Products

(pp.98-101).
A: Reading questions. Hand in at beginning of class, Thu. 23 Mar.
1. The text claims near the bottom of p. 98 that "[t]he norm is not linear on R^n." Verify this claim. [Hint: Define a function T: R^n --> R^n by N(x)= ||x||, and show N is not linear.] How does this claim relate to the introduction of inner products?
2. Provide a little more explanation for the claim near the bottom of p. 99, "The equation above thus suggests that the inner product of w = (w_1, ... ,w_n) element-of C^n with z should equal
w_1 bar{z}_1 + ... + w_n bar{z}_n."
[Note: I am typesetting the complex conjugate of complex number z as bar{z}.]
3. Match the properties of the dot product described at the bottom of p. 98 to the five properties listed at the top of p. 100 that define an inner product.
4. Provide justification for each step in the derivation, on p. 101, that < u, v+w > = < u, v > + < u, w >. Note that some of these will be properties of inner products, and others will be properties of complex conjugates (see p. 69).
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 28 Mar.
• Verify 6.1 and 6.2 define inner products, as claimed.
C: Main exercises.
• none

### Norms

(pp. 102-106).
A: Reading questions. Hand in at beginning of class, Tue. 28 Mar.
1. Provide justification for each step in the derivations, on p. 102 and p. 105, respectively, that ||av|| = |a| ||v||, and ||u+v|| <= ||u|| + ||v||. Note that some of these will be properties of inner products, and others will be properties of complex conjugates (see p. 69).
2. Verify the claim, below equation 6.5, that if v not= 0, this equation writes u as a scalar multiple of v plus a vector orthogonal to v.
3. Directly verify the Cauchy-Schwarz inequality (6.6) for the following pairs of vectors:
• (3,1,4) and (2,7,1) in R^3, with inner product 6.1; and
• x^2 and 7x-2 in P_2(R), with inner product 6.2.
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 30 Mar.
• Justify the steps of 6.14.
• Ch. 6: 4.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 6.
• Ch. 6: 1, 5.

### Orthnormal Bases

(pp. 106-110). We will only be considering material in this section up to and including the proof of Corollary 6.25.
A: Reading questions. Hand in at beginning of class, Thu. 30 Mar.
1. Verify the list of four vectors in R^4 given in the middle of p. 107 is indeed orthonormal.
2. Demonstrate Theorem 6.17 with V=R^4, orthonormal basis (e_1,e_2,e_3,e_4) given by the list in question 1 above, and v=(4,3,2,6).
3. Try to read the proof of the Gram-Schmidt theorem (6.20) without worrying too much about the precise algebraic details of equation 6.23 or the calculation at the top of p. 109. The third sentence of the proof says, "We will choose e_2, ... ,e_m inductively ... ". What, in your own words, does that mean in this case?
4. Near the top of p. 108, the text asks, "does P_m(F), with inner product [given by 6.2] have an orthonormal basis?" Answer this question, and explain your answer. [Note: you do not have to produce such a basis, just decide whether or not it exists.]
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 4 Apr.
• Ch. 6: 10.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 13.
• Ch. 6: 13.
• Ch. 6: 11. [Grad students only]

### Orthogonal Projections and Minimization Problems

(pp. 111-116).
A: Reading questions. Hand in at beginning of class, Tue. 4 Apr.
1. Find U^perp for U = span((9,1,5)) in V = R^3. Describe U^perp geometrically in this case. [Note I am typesetting the "perpendicular" symbol by "perp".]
2. Verify Theorem 6.29 in the case of question 1 above.
3. Find P_U v for v=(1,2,3) and U= span((9,1,5)) in V=R^3.
4. In the example starting on p. 114, approximating sin x by a 5th-degree polynomial, explain how integral_{-pi}^{pi} | sin x - u(x) |^2 dx is minimized using the inner product 6.39 and Proposition 6.36.
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 6 Apr.
• Verify the five properties of P_U listed near the top of p. 113.
• Ch. 6: 19, 21.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 13.
• Ch. 6: 20, 22.

### Generalized Eigenvectors (part I)

(pp. 164-167).
A: Reading questions. Hand in at beginning of class, Thu. 6 Apr.
1. The text states, in the middle of p. 164, that the operator in 5.19 "does not have enough eigenvectors for 8.2 to hold." Explain carefully what that means in this case.
2. Show how the example at the top of p. 165 matches the equation near the bottom of p. 164, V = null(T - lambda_1 I)^(dim V) "oplus" ... "oplus" null(T - lambda_m I)^(dim V).
3. The text claims, in the margin of p. 165, that "if (T - lambda I)^j is not injective for some positive integer j, then T - lambda I is not injective ...". Verify this claim.
4. Verify Proposition 8.5 for the linear operator T element-of L(F^4) given by T(z_1,z_2,z_3,z_4)=(z_1,z_3,z_4,0).
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 11 Apr.
• Ch. 8: 1.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 20.
• Ch. 8: 2.

### Generalized Eigenvectors (part II)

(pp. 167-168).
A: Reading questions. Hand in at beginning of class, Tue. 11 Apr.
1. Verify the claim at the top of p. 167 that the operator N element-of L(F^4) defined by N(z_1,z_2,z_3,z_4)=(z_3,z_4,0,0) satisfies N^2=0.
2. Find a linear operator in L(F^4) that is not nilpotent, and show it is not nilpotent.
3. Explain more carefully the following claim made at the beginning of the proof of Corollary 8.8: "Because N is nilpotent, every vector in V is a generalized eigenvector corresponding to the eigenvalue 0."
4. Verify both Proposition 8.9 and the displayed equation above it, V= range T^0 superset range T^1 superset ... superset range T^k superset range T^(k+1), for the linear operator T element-of L(F^4) given by T(z_1,z_2,z_3,z_4)=(z_1,z_3,z_4,0).
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 13 Apr.
• Ch. 8: 5, 6.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 20.
• Ch. 8: 8.
• Ch. 8: 9. [Grad students only]

### The Characteristic Polynomial (part I)

(pp. 168-171). [Part I covers through the end of the proof of Theorem 8.10.]
A: Reading questions. Hand in at beginning of class, Thu. 13 Apr.
1. Answer the question posed in the middle of p. 168, "Could the number of times that a particular eigenvalue is repeated depend on which basis of V we choose?"
2. Demonstrate Theorem 8.10 on on the 4-by-4 upper triangular matrix near the top of p. 83. In other words, show that dim null (T - lambda I)^(dim V) is 2 for lambda=6, since 6 appears twice on the diagonal, and is 1 [note typo from handout--"4" should be replaced by "1" here!] for lambda=7,8, since 7 and 8 each appear once on the diagonal. Note that the basis here is the standard basis.
3. Demonstrate the claim, made in the margin of p. 168, that if T has a diagonal matrix A with respect to some basis, then lambda appears on the diagonal of A precisely dim null (T - lambda I) times [note typo in handout, missing "I"], on the linear operator T element-of L(F^3) defined by T(z_1,z_2,z_3)=(4z_1,4z_2,5z_3) on p. 88. Note that the basis here is the standard basis. Why is this claim a special case of Theorem 8.10?
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 18 Apr.
• Ch. 8: 10.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 27.
• Ch. 8: 11.

### The Characteristic Polynomial (part II)

(pp. 171-173).
A: Reading questions. Hand in at beginning of class, Tue. 18 Apr.
1. Verify, using the definition of multiplicity, the following claim made on p. 171, of the linear operator T defined in equation 8.16: "0 is an eigenvalue of T with multiplicity 2 [and] that 5 is an eigenvalue of T with multiplicity 1".
2. Where does the proof of Proposition 8.18 use the assumption that V is a complex vector space?
3. Fill in the missing details of the claim in the middle of p. 172, that the characteristic polynomial of the operator defined by equation 8.16 equals z^2(z-5).
4. Verify, directly, that the operator T defined in equation 8.16 satisfies the Cayley-Hamilton theorem. [Hint: This is easier to verify by first converting T to a matrix.]
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 20 Apr.
• Ch. 8: 14.
C: Main exercises. Hand in at beginning of class, Thu. Apr. 27.
• Ch. 8: 15.
• Ch. 8: 13. [Grad students only]

### Decomposition of an Operator (part I)

(pp. 173-175). [Part I covers through the end of the proof of Corollary 8.25.]
A: Reading questions. Hand in at beginning of class, Thu. 20 Apr.
1. Justify each equality in the displayed string of equalities in the center of the proof of Proposition 8.22. Any step you can't justify, demonstrate with a small example (say, where p is a quadratic polynomial, and T is an operator on F^2).
2. Theorem 8.23 is our major goal of this chapter. Demonstrate all three parts of Theorem 8.23 on the linear operator T defined in equation 8.16. The proof of Theorem 8.23 may give some hints, though you don't need to understand all the details of this proof to answer this question. It may also help to use the matrix representation of T and other operators.
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 25 Apr.
• Verify Corollary 8.25 on the 4-by-4 matrix on p. 83 (find a basis of generalized eigenvectors).
C: Main exercises.
• None. (But see part II of this section below.)

### Decomposition of an Operator (part II)

(pp. 175-176).
A: Reading questions. Hand in at beginning of class, Tue. 25 Apr.
1. Prove the converse of Lemma 8.26, that any matrix of the form 8.27 is nilpotent. Find a good example of your result, on a nontrivial 4-by-4 nilpotent matrix. (Here, "good" and "non-trivial" mean with as few zero's as you can.)
2. The first step of the proof of Lemma 8.26 is to choose a basis of null N and then extend it to a basis of null N^2. Provide the details of why we can extend the basis of null N to a basis of null N^2.
3. Prove the converse of Theorem 8.28, that the eigenvalues of any block diagonal matrix whose blocks have the form 8.29 are lambda_1, ... ,lambda_m. Construct a good example of a 7-by-7 matrix of this form, with eigenvalues 3, 0, and -1. (Here, "good" again means with as few zero's as possible.)
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 27 Apr.
• Ch. 8: 12.
C: Main exercises. Hand in at beginning of class, Thu. May 4.
• Ch. 8: 17.

### The Minimal Polynomial (part I)

(pp. 179-180). [Part I covers through the end of the proof of Theorem 8.34.]
A: Reading questions. Hand in at beginning of class, Thu. 27 Apr.
1. Compare and contrast the argument for the existence of the minimal polynomial in the second paragraph of this section with the proof of Theorem 5.10.
2. Now let's focus on the uniqueness of the minimal polynomial. Provide more explicit algebraic details of the claim made just below the middle of p. 179 that "[t]he choice of scalars a_0, a_1, a_2, ... ,a_(m-1) element-of F is unique". (The text gives a summary of what to do in the parenthetical comment at the end of that sentence; make that summary more explicit, including writing some equations.)
3. Verify the claim made near the bottom of p. 179 that the minimal polynomial of the operator on F^2 whose matrix equals [the matrix with first row (4, 1) and second row (0, 5)] is 20 - 9z + z^2.
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 2 May.
• Ch. 8: 21, 22, 23.
C: Main exercises.
• None.

### The Minimal Polynomial (part II)

(pp. 180-182).
A: Reading questions. Hand in at beginning of class, Tue. 2 May.
1. Is there any relation between the minimal polynomial and the characteristic polynomial?
2. What, if anything, does Theorem 8.36 say about the multiplicity of each root of the minimal polynomial?
3. Show the details of how the text found the minimal polynomial of the linear operator whose matrix is given by 8.37 on the top of p. 182 (you may use technology such as a calculator or computer). How does this help compute the eigenvalues of the linear operator?
4. What will you do with all the time you have, now that there are no more reading questions to answer?
B: Warmup exercises. For you to present in class. Due by the end of class Thu. 4 May.
• Ch. 8: 26, 27.
C: Main exercises.
• None.