# Math 4326 Linear Algebra Homework

## Dr. Duval

### Complex NumbersDefinition of Vector Space

(pp. 1-10)
A: Reading questions. Hand in Thu. 17 Jan., or earlier.
1. Verify, using properties of real numbers, and that i^2=-1, that complex numbers satisfy the distributive property.
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. 17 Jan.
• Verify distributivity in P(F).
• Ch. 1: 2.
C: Main exercises. due Fri. 25 Jan., 2pm
• 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. Due in class Thu., 17 Jan., with an automatic extension to Friday (hardcopy) or Monday (electronic)
1. In Proposition 1.3, why do we "[s]uppose that w and w' are additive inverses of v"?
2. In Proposition 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 one other condition on p. 9, and show how the conditions on p. 13 lead to it
5. Check that the second example at the bottom of p. 13 is indeed a subspace, as claimed in the text.
B: Warmup exercises. For you to present in class. Due in class Tue. 22 Jan.
• Ch. 1: 3, 5, 6.
C: Main exercises. due Fri. 1 Feb., 2pm.
• Ch. 1: 8.

### Sums and Direct Sums

(pp. 14-18).
A: Reading questions. Due by 5pm, Wed. 23 Jan., if possible; beginning of class Thu. 24 Jan., otherwise.
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. 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.]
6. 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. 24 Jan.
• Ch. 1: 10, 11, 13.
C: Main exercises. due Fri. 1 Feb., 2pm.
• 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. Why should the span of an empty list be {0} [the vector space whose only vector is the 0 vector]?
3. Verify that, if some vectors are removed from a linearly independent list, then the remaining list is also linearly independent.
4. Why is the empty list linearly independent?
5. 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.]
6. Demonstrate the multistep process described in the proof of Theorem 2.6 on the linearly independent list ((1,1,1), (1,2,0)) and the linearly dependent list ((2,3,1), (1,-1,2), (7,3,8)).
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. due Fri. 8 Feb., 2pm.
• Ch. 2: 3.
• Ch. 2: 5. [Grad students only]

### Bases

(pp. 27-31).
A: Reading questions. Due by 2pm, Wed., 30 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., 31 Jan.
• Ch. 2: 8, 9.
C: Main exercises. due Fri. 8 Feb., 2pm.
• Ch. 2: 10.

### Dimension

(pp. 31-34).
A: Reading questions. Due by 2pm, Mon., 4 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. Verify that Theorem 2.18 works when U_1 is the xy-plane, and U_2 is the yz-plane, in R^3.
5. 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., 5 Feb.
• Ch. 2: 11, 12, 13.
C: Main exercises. due Fri. 15 Feb., 2pm.
• Ch. 2: 16.
• Ch. 2: 17. [Grad students only]

### Definitions and Examples

(pp. 38-41).
A: Reading questions. Due by 2pm, Wed., 6 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. 7 Feb.
• Ch. 3: 1, 2.
C: Main exercises. due Fri. 15 Feb., 2pm.
• Ch. 3: 3.

### Null Spaces and Ranges (part I)

(pp. 41-44).
A: Reading questions. Due by 2pm, Mon., 11 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. Identify, as precisely as you can, when we use the linearity of T in the proofs of Propositions 3.1, 3.2, and 3.3. [Pinpoint the exact equations and statements that depend on linearity, and which part of the definition of linearity that is used in each case.]
4. 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.
B: Warmup exercises. For you to present in class. Due by the end of class Tue. 12 Feb.
• Ch. 3: 5, 6.
C: Main exercises. due Fri. 22 Feb., 5pm
• Ch. 3: 7.

### Null Spaces and Ranges (part II)

(pp. 45-47).
A: Reading questions. Due by 2pm, Wed., 13 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., 14 Feb.
• Ch. 3: 9, 12.
C: Main exercises. due Fri. 22 Feb., 5pm
• Ch. 3: 13.
• Ch. 3: 11. [Grad students only]

### The Matrix of a Linear Map (part I)

(pp. 48-50).
A: Reading questions. Due by 2pm, Mon., 18 Feb.
1. In the middle of p. 49, the textbook offers an example corresponding to a 3-by-2 matrix. Make up your own example corresponding to a 2-by-4 matrix, with no 0 entries, and all entries being different. Now explain your example just as carefully as the textbook explains its example. You may use the textbook's example as a template for your own.
2. Verify equations 3.9 and 3.10.
3. Near the bottom of p. 50, the text claims that Mat(m, n, F) is a vector space, and asks you to verify this. Though you may want to check all the properties of a vector space (see p. 9) for yourself, please just turn in a verification of the first distributive property on p. 9.
B: Warmup exercises. For you to present in class. Due by the end of class Tue., 19 Feb.
• Differentiation: 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'. If doing this for arbitrary m is too hard, try the special case where m = 5 first.
• Multiplication by x^2: Repeat for T element-of L(P_m(F), P_(m+2)(F)), the multiplication by x^2 linear map, i.e., T(p(x))= x^2p(x).
C: Main exercises. due Fri. 29 Feb., 2pm
• Integration: Modify the integration map on the top of p. 39 so that the domain of T is P_m instead of P. Carefully construct M(T) for an appropriate choice of bases. Clearly state what bases you are using.

### The Matrix of a Linear Map (part II)

(pp. 51-53).
A: Reading questions. Due by 2pm, Wed., 20 Feb.
1. In the long string of equalities in the middle of p. 51, what are u_k, v_r, and w_j?
2. 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?
3. Why do we need a proof for Proposition 3.14? Why can't we just use equation 3.11?
B: Warmup exercises. For you to present in class. Due by the end of class Thu., 21 Feb.
• Ch. 3: 17, 18, 19.
C: Main exercises. due Fri. 29 Feb., 2pm
• Ch. 3: 21.

### Invertibility (part I)

(pp. 53-55).
A: Reading questions. Due by 2pm, Mon., 25 Feb.
1. How many inverses can a linear transformation have?
2. When, if ever, in the proof of Proposition 3.17 do we use the linearity of T or of any other map?
3. In the first half of the proof of Theorem 3.18, it is claimed, "Because T is invertible, we have null T = {0} and range T = W." Why is this implication true?
4. In the second half of the proof of Theorem 3.18, an invertible linear map T is defined (different from the T in the first half of the proof). 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.

B: Warmup exercises. For you to present in class. Due by the end of class Tue., 26 Feb.
• Ch. 3: 22.
C: Main exercises. due Fri. 14 Mar., 2pm
• Ch. 3: 23.
• Ch. 3: 24. [Grad students only]

### Invertibility (part II)

(pp. 56-58).
A: Reading questions. Due by 2pm, Wed., 27 Feb.
1. 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.]
2. An easy corollary to Proposition 3.19 is that two vector spaces are isomorphic. Which ones, and why?
3. Why is Theorem 3.21 remarkable?
4. 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. 28 Feb.
• Ch. 3: 20.
C: Main exercises. due Fri. 14 Mar., 2pm
• Ch. 3: 26.

### 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. Due by 2pm, Mon., 3 Mar.
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! Pick a polynomial that will make your job easier.]
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)"?
4. Why might your answer to question 2 above seem to contradict Corollary 4.8? Why doesn't it actually give a contradiction?
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. 4 Mar.
• Ch. 4: 1, 5.
C: Main exercises.
• none

### Invariant Subspaces

(pp. 76-79).
A: Reading questions. Due by 2pm, Mon., 10 Mar.
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 Tue. 11 Mar.
• Ch. 5: 1, 5, 6.
C: Main exercises. due Tue. 25 Mar., 5pm note change
• Ch. 5: 2. [Note: For some reason, I didn't announce this in class, so it will count for extra credit!
• Ch. 5: 4. [Grad students only]

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

(pp. 80-83).
A: Reading questions. Due by 2pm, Wed., 12 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 Thu. 13 Mar.
• Ch. 5: 9, 10.
C: Main exercises. due Tue. 25 Mar., 5pm note change
• Ch. 5: 8, 14.

### Upper-Triangular Matrices (part II)

(pp. 83-87).
A: Reading questions. Due by 2pm, Mon., 17 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. 18 Mar.
• Ch. 5: 18, 19.
C: Main exercises. due Fri. 4 Apr., 2pm
• Ch. 5: 17.

### Diagonal Matrices

(pp. 87-90).
A: Reading questions. Due by 2pm, Wed., 19 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 that 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. 20 Mar.
• Ch. 5: 11.
C: Main exercises. due Fri. 4 Apr., 2pm
• Ch. 5: 20.
• Ch. 5: 21. [Grad students only.]

### Inner Products

(pp.98-101).
A: Reading questions. Due by 8:15am, Tue., 1 Apr. note change!
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 N: R^n --> R 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. 1 Apr.
• Verify 6.1 and 6.2 define inner products, as claimed.
C: Main exercises.
• None, but see homework for Norms below.

### Norms

(pp. 102-106).
A: Reading questions. Due by 2pm, Wed., 2 Apr.
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. 3 Apr.
• Justify the steps of 6.14.
• Ch. 6: 4.
C: Main exercises. due Fri. 11 Apr., 2pm
• 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. Due by 2pm, Mon., 7 Apr.
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. 8 Apr.
• Ch. 6: 10.
C: Main exercises. due Fri. 18 Apr., 2pm
• Ch. 6: 13.
• Ch. 6: 11. [Grad students only]

### Orthogonal Projections and Minimization Problems

(pp. 111-116).
A: Reading questions. Due by 2pm, Wed., 9 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 Tue. 15 Apr. Note change in due date
• Verify the five properties of P_U listed near the top of p. 113.
• Ch. 6: 19, 21.
C: Main exercises (part 1). due Fri. 18 Apr., 2pm
• Ch. 6: 15.
C: Main exercises (part 2). due Fri. 25 Apr., 2pm
• Ch. 6: 20, 22.

### Generalized Eigenvectors (part I)

(pp. 164-167).
A: Reading questions. Due by 2pm, Wed., 16 Apr. Note change in due date
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 Thu. 17 Apr. Note change in due date
• Ch. 8: 1.
C: Main exercises. due Fri. 25 Apr., 2pm
• Ch. 8: 2.

### Generalized Eigenvectors (part II)

(pp. 167-168).
A: Reading questions. Due by 2pm, Mon., 21 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 Tue. 22 Apr.
• Ch. 8: 5, 6.
C: Main exercises. Due at the beginning of class, 1:30pm, Thu., 1 May.
• 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. Due by 2pm, Wed., 23 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 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, 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 Thu. 24 Apr.
• Ch. 8: 10.
C: Main exercises. Due at the beginning of class, 1:30pm, Thu., 1 May.
• Ch. 8: 11.

### The Characteristic Polynomial (part II)

(pp. 171-173).
A: Reading questions. Due by 2pm, Mon., 28 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.]
5. 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 Tue. 29 Apr.
• Ch. 8: 14.
C: Main exercises.
• None.