Homework for MATH 4330 in Fall 2022


return to Fall 2022 MATH 4330 webpage

Recommended textbook
(not required):		 Linear Algebra (Pure & Applied Undergraduate Texts) by Michael E. Taylor,
American Math. Soc., 2020.
See our course’s Canvas portal for access options.
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  • There might be more than one correct answer for any given question.
  • Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative and subject to change.
  • If you notice some questions are in a nonincreasing order, then it means that the order is recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
  • Any optional exercises are for those students who wish to explore some of the ideas in more depth – those questions are NOT game for the tests.
  • Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture – try it!
  • If any of the mathematics below does not display properly (e.g., an error message received), compare with this file.
  • Recall that our test dates are: Sept 21, Nov 2 and Dec 12, and the due dates of the online quizzes will be announced in class and/or in the homework.
  • LAST REVISION: 12/03/22.

Aug 22

Check your Canvas notifications to check you can receive Canvas announcements.
Attendance will be noted, starting today.
Read course syllabus carefully. Make a note of the test dates in your calendar.
Review course website and repeat frequently during the semester.
Review the course’s Canvas portal.

Read this study tip and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively.

Read your lecture notes (meaning the notes you should have taken during lecture) and skim/read Section 2.1.
Do Sec 2.1: 2.

Recall from today’s lecture that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Aug 24

Read your lecture notes, skim/read Section 2.1 and read page 5 of the instructor’s lecture notes from Fall 2021 (see our Canvas portal) and
do Sec 2.1: 11, 8 and
H0. Let A = ; diagonalize A and use the diagonal matrix to compute A100.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.

Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Aug 29

Read your lecture notes and do Sec 2.2: 1, 2 (but not minimal polynomial part). (Note that Aj in #2 is referring to the matrices in Question 2.2.1.)
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

The information sheet for Quiz 1 is posted in Canvas.

Aug 31

Read your lecture notes and do
Sec 2.2: 2 (minimal polynomial part only and note that Aj in #2 is referring to the matrices in Question 2.2.1), and
H1. Repeat Questions 2.2.1 and 2.2.2 for the matrix A4 , where

Sec 2.2: 3.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.

Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

See the information sheet for Quiz 1 in Canvas.

Sep 05

Labor Day Holiday (no lecture)
Sep 07 Read your lecture notes and do
H2. Let A5 be the matrix

(a) Use the eigenvalues of A5 to show that A5 is nilpotent.
 (b) Determine the smallest positive exponent k such that A5k = 0.

H3. (a) Determine which of the matrices A1, …, A4 from Questions 2.2.1 and H1 are nilpotent.
(b) For each matrix in (a) that is nilpotent, determine the smallest positive exponent k such that Aik = 0.
Sec 2.3: 2, 3 (assume the notation in question 3 is referring to question 2).

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Quiz 1 opens today; it is due Sept 18.
Sep 12

Read your lecture notes and do
Sec 2.3: 1 and
watch the 61-minute video on companion matrices and rational canonical form (use the link provided or access it in the Modules section of our Canvas portal) and then do
H4. (a) Compute the companion matrix for the polynomial found in Question 2.3.1.
(b) Compute the companion matrix of the characteristic polynomial of the matrix given in Question 2.1.11.
H5. (a) Suppose A and B are similar matrices. Show that Ak = 0 iff Bk = 0, where k ∊ ℕ.
(b) Find the characteristic polynomial of the matrix A4 given in H1.
 (c) Compute the companion matrix C of the polynomial found in (b).
 (d) Compute A42 and C2.
 (e) Use (a) and (d) to show that A4 and C are NOT similar matrices.
 (H5(e) motivates using rational canonical form instead of C.)
H6. Find the rational canonical form of each of the following matrices:
 (a) the matrix A4 given in H1
 (b)
(c)
(d) the second matrix and the third matrix in Question 2.4.1.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 1 ; it is due Sept 18.
Sep 14

Read your lecture notes.
Strongly recommended: read the file on JCF used in Math 5333.
Do
Sec 2.4: 1 and

H7. Find the minimal polynomial of each matrix in Question 2.4.1 and factor the polynomials.

H8. Let A denote the matrix A4 given in H1.
 (a) Find the Jordan canonical (normal) form J of A.
 (b) Find an invertible matrix P such that P-1 A P = J.
 (c) Factor the minimal polynomial of A.
H9. Let A be the matrix

(a) Find the Jordan canonical (normal) form J of A.
 (b) Find an invertible matrix P such that P-1 A P = J.
 (c) Find the minimal polynomial of A and factor it.
H10. Let A be the matrix

(a) Find the Jordan canonical (normal) form J of A.
 (b) Find an invertible matrix P such that P-1 A P = J.
 (c) Find the minimal polynomial of A and factor it.
H11.Let A be the matrix
(a) Find the Jordan canonical (normal) form J of A.
 (b) Find an invertible matrix P such that P-1 A P = J.
 (c) Find the minimal polynomial of A and factor it.

H12.

What do you notice about the (factored) minimal polynomials for the matrices in H7-H11 and their corresponding Jordan canonical (normal) forms?

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 1 ; it is due Sept 18.

Test 1 will be 1 week from today; check Canvas for an information sheet.
Sep 19

Read your lecture notes and do
H13. Find a vector v ∊ ℂ2 \ ℝ2 with the property that v · v ∉ ℝ (so the dot product is NOT an inner product on ℂ2 ).
H14. Show that the euclidean inner product on ℂ2 is an inner product on ℂ2.
H15. Show that properties (c) & (e) from the definition of inner product given in lecture imply that an inner product
satisfies additivity on the right: (w, u+v) = (w, u) + (w, v) for all u, v, w that belong to the IPS.
H16. Show that properties (d) & (e) from the definition of inner product given in lecture imply that an inner product
satisfies conjugate homogeneity on the right: for all u, v that belong to the IPS and for
all α ∊ 𝔽.
H17. Consider ℝn with the dot product. Let v, w ∊ ℝn and suppose that ||v|| = 1 = ||w||. Show that v – w is orthogonal
to v + w.
H18. Let V denote an IPS and assume that the parallelogram inequality holds; namely,
||u+v||2 + ||u – v||2 = 2 ( ||u||2 + ||v||2 ) for all u, v ∊ V.
If u, v ∊ V satisfy: ||u||=3, ||u+v|| = 4, ||u – v|| = 6, what does ||v|| equal?

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.

Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Test 1 will be on Wednesday; check Canvas for an information sheet.
Sep 21 Test 1 today; check Canvas for an information sheet.
Sep 26

Read your lecture notes and look over the solutions to Test 1 in the Assignments section of our Canvas portal, and do

H19. Assume, without proof, that the pairing for all ai, bi ∊ ℝ, defines an
inner product on the linear space V of all 2 x 2 matrices with entries in ℝ. Use the Gram-Schmidt formula
(GSOP) on the matrices

to produce an orthonormal set of elements in V.
H20. Suppose 𝔽 = ℝ and let V denote the space of all polynomials in the variable x of degree at most 2 with
coefficients in ℝ. Suppose V is an IPS with inner product ( , ) that satisfies:
(1, 1) = 1/3, (1, x) = ¼, (1, x2) = (x, x) = 1/5, (x, x2) = 1/6, (x2, x2) = 1/7.
Let u1 = ∊ V and u2 = (4x – 3) ∊ V.
(a) Justify that u1 and u2 are orthonormal.
(b) Use the Gram-Schmidt formula (GSOP) to find 0 ≠ u3 ∊ V such that u1, u2 and u3 are orthonormal.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.
Sep 28

Read your lecture notes and do
H21. Suppose 𝔽 = ℂ and let T : ℂ3 → ℂ2 be defined by

(a) Using the euclidean inner product on ℂ3 and ℂ2, find a formula for T*: ℂ2 → ℂ3.
 (b) Find the matrix of T and the matrix of T* with respect to the standard bases of ℂ2 and ℂ3.
 (c) Compare your matrices in (b) with the first remark in the lecture on Section 3.2.

H22. Suppose 𝔽 = ℝ and let V denote the vector space of polynomials in the variable x of degree at most 1 with
 coefficients in ℝ and let T = d/dx : V→V. Using the IP for all f, g ∊ V, find
a formula for T*.

H23. Let V denote a finite-dimensional IPS. Suppose u1, u2 ∊ V. Let T : V → V be the linear map defined by
T(v) = (v, u1) u2 for all v ∊ V. Find a formula for T*.

H24. Let V denote a finite-dimensional IPS. Let I denote the identity operator on V; that is I(v) = v for all v ∊ V.
Justify that I* = I.

H25. Let T : ℝ2 → ℝ be the linear map Using the dot product for the
IP on ℝ2, find a vector u ∊ ℝ2 such that (v, u) = T(v) for all v ∊ ℝ2.

H26. Suppose 𝔽 = ℝ and let V denote the space of all polynomials in the variable x of degree at most 2 with
coefficients in ℝ. View V as an IPS via for all f, g ∊ V. Let T : V → ℝ be defined
by for all f ∊ V (where denotes the derivative of f with respect to x).
Find u ∊ V such that ( f , u ) = T( f ) for all f ∊ V.
(Hint: find an orthonormal basis of V (e.g., see the lecture notes on Section 3.1) and see the proof of the
claim in the lecture on Section 3.2.)

H27. (Optional) Let V denote the vector space of all infinite sequences v = (v1, v2, .…) where vi ∊ 𝔽 for
all i and vi = 0 for all but finitely many i. Let ei = (0,…, 0, 1, 0, ….) where the 1 occurs in the i’th position.
View V as an IPS by using for all v, y ∊ V, where v = (v1, v2, .…), y = (y1, y2, .…).
Let T : V → V denote the linear map for all
v = (v1, v2, .…) ∊ V. (Optional: prove that T is a linear map.)
(a) Verify that for all n ∊ ℕ.
(b) Justify that the adjoint of T does not exist. (Hint: suppose T* does exist and show that
T*(en) = (z1, z2, ….) where zi ∊ 𝔽 for all i, but zi is nonzero for infinitely many i (so T*(en) ∉ V).)

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.
Oct 03

Read your lecture notes and do
H28. Show that AtA-AAt is a symmetric matrix for all nxn matrices A (recall At denotes the transpose of A).
H29. Let V denote a finite-dimensional IPS. Show that T* ○ T – T ○ T* is a self-adjoint linear map for all linear maps T : V→V.
Sec 3.3: 1 (assume the IP is the dot product).
H30. Let V denote a finite-dimensional IPS. A linear map T : V→V such that T* ○ T = T ○ T* is called normal.
 Show that all self-adjoint linear maps are normal.
H31. For each of the following, assume V is endowed with the euclidean inner product.
In each case, determine if the linear map T is self-adjoint or normal (see previous question) or neither.
 (a) 𝔽 = ℝ, V = ℝ2, for all x, y ∊ ℝ.

(b) 𝔽 = ℝ, V = ℝ3, for all x, y , z ∊ ℝ.

(c) 𝔽 = ℂ, V = ℂ2 , for all x, y ∊ ℂ, where i2 = –1.

(d) 𝔽 = ℂ, V = ℂ2 , for all x, y, z ∊ ℂ, where i2 = –1.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.
Oct 05

Read your lecture notes and do
H32. Suppose 𝔽 = ℝ and let V = span{ 1, cosx, sinx }, and let D = d/dx.
Assume, without proof, that D : V→V is linear and that for all f, g ∊ V defines an IP
on V and that f(π) g(π) = f(-π) g(-π) for all f, g ∊ V.
(a) Justify that D is skew-adjoint. (Hint: integration by parts.)
(b) Determine whether or not D is normal (see homework from the last lecture for the definition).
(c) Determine whether or not D○D is normal or self-adjoint.
Sec 3.3: 2 (assume the IP is the dot product) and
H33. Suppose 𝔽 = ℝ and use the dot product on ℝn.
Show that AB-BA is skew-adjoint for all skew-adjoint nxn matrices A and B.
H34. Let V and W denote finite-dimensional inner-product spaces. Justify that
T* ○ T : V→V is positive semidefinite and that
T ○ T* : W→W is positive semidefinite
for all linear maps T : V→W.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.
Oct 10

Read your lecture notes and do
H35. Suppose 𝔽 = ℝ and let V denote a finite-dimensional IPS and T : V→V a linear map.
We define b : V x V→ℝ by b(x, y) = (x, T(y)) for all x, y ∊ V.
(a) Justify that b is a bilinear form.
(b) Justify that b is symmetric if and only if T is self-adjoint.
H36. Suppose 𝔽 = ℝ. For each of the given quadratic forms Q, find a symmetric bilinear form b such that
b(x, x) = Q(x) for all x ∊ V, for a suitable V in each case.
(a) Q : ℝ2 → ℝ where for all x, y ∊ ℝ.

(b) Q : ℝ3 → ℝ where for all x, y, z ∊ ℝ.

H37. Suppose 𝔽 = ℝ.
(a) Determine the quadratic form corresponding to the matrix .

(b) Find the symmetric matrix corresponding to the quadratic form
for all x1, …, x4 ∊ ℝ.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 2 ; it is due Oct 24.
Oct 12 Read your lecture notes and do
H38. Suppose 𝔽 = ℝ and let

(a) Verify that A is orthogonal.
(b) Verify that the characteristic polynomial of A is (x+1)(x2+1) = (x+1)(x+i)(x–i), where i2 = –1.
(c) Find the eigenspaces of A and verify that they are orthogonal with respect to the euclidean IP on ℂ3.
(d) Find the new matrix that represents the linear map given by A with respect to the (reordered) standard
basis B = {e2, e1, e3} of ℝ3.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 2 ; it is due Oct 24.
Oct 17

Read your lecture notes and do
H38 (continued — continuing from Oct 12 above)
(e) Compare the matrix in (d) with the last theorem of the lecture on Section 3.4.
(f) Compare the basis vectors in B with the eigenvectors found in (c) (consider real and imaginary parts
of the entries).

Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 2 ; it is due Oct 24.
Oct 19 Read your lecture notes, especially the examples of polar decomposition (PD) and singular-value decomposition (SVD), and read the example in the recommended textbook that appears right after Proposition 3.6.7, and do

H39. Find the polar decomposition of the matrix
H40. Find the polar decomposition of the matrix A, where A is the third matrix in Exercise 1 of Section 3.6.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 2 ; it is due Oct 24.
Oct 24

Read your lecture notes and do
H41. Find the SVD of the matrix
H42. Find the SVD of the third matrix given in Exercise 2 of Section 3.6.
H43. Find the SVD of the matrix
H44. Let A = .
(a) Find the JCF of A.
(b) Find the SVD, A = UDVt, of A.
(c) Solve the SVD in (b) for D and, compare D with the matrix found in (a).
H45. Let A = .
(a) Find the JCF of A.
(b) Find the SVD, A = UDVt, of A.
(c) Solve the SVD in (b) for D and, compare D with the matrix found in (a).

The last 2 questions highlight that even if P-1AP is not diagonal for any invertible matrix P, a diagonal matrix can be associated to A if the left matrix (U) and the right matrix (V) are not as tightly related as UVt = I (& the SVD takes this further by having U and V be isometries).

H46. (optional) Suppose A is an nxn matrix with SVD given by A = UDV*.
Show that the PD of A is A = K , where K = UV* and = VDV*.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 2 ; it is due today.
Oct 26 Read your lecture notes, and do
H47. Find eIt where I is the 2×2 identity matrix.
Sec 3.7: 16.

 

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Test 2 will be 1 week from today; check Canvas for an information sheet.
Oct 31 Read your lecture notes, and do
H48. Suppose 𝔽 = ℝ and let V denote ℝ3 with the dot product for the IP. Let U denote the plane in V spanned
by the vectors

H49. Suppose 𝔽 = ℝ. In ℝ4, let Find u ∊ U such that is as small as possible.
If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Test 2 will be on Wednesday; check Canvas for an information sheet.
Nov 02 Test 2 today; check Canvas for an information sheet.
Nov 07 Read your lecture notes and look over the solutions to Test 2 in the Assignments section of our Canvas portal, and do
H50. Suppose 𝔽 = ℝ and let V denote the linear space of all polynomials of degree at most three in the variable x
with coefficients in ℝ. Find a polynomial p in V such that
 is as small as possible.

 

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.
Nov 09 Read your lecture notes and do
H51. For each of the following, discuss whether or not the given function T on a linear (vector) space V determines a linear functional.

(a) V = ℝ2, for all x, y ∊ ℝ.
(b) V = ℝ3, for all x, y, z ∊ ℝ.
(c) V = the space of all 2×2 matrices with entries in ℝ, T(M) = the sum of the diagonal entries of M, for all
M ∊ V.
(d) V = the space of all 2×2 matrices with entries in ℝ, T(M) = the 22-entry of M, for all M ∊ V.
(e) V = the space of all 2×2 matrices with entries in ℝ, T(M) = the determinant of M, for all M ∊ V.
H52. Let B denote the basis of ℝ3. Find the dual basis of relative to B.

H53. Suppose n ∊ ℕ and let B denote the basis {1, x, …, xn} of the linear (vector) space V of all polynomials in x,
with coefficients in ℝ, of degree at most n. Let for all p ∊ V, where p(0) = p ,
etc, and 0! = 1. Show that { } is the dual basis of relative to B.
H54. Suppose n ∊ ℕ and let B denote the basis {1, x, …, xn} of the linear (vector) space V of all polynomials in x,
with coefficients in ℝ, of degree at most n.
(a) If n = 2, show that {1, x-7, (x-7)2} is a basis of V.
(b) If n = 2, find the dual basis of relative to the basis in (a). (Hint: consider H53, but tweak it.)
H55. Let V denote a linear (vector) space (possibly of infinite dimension).
(a) Show that T1 + T2 is a linear functional for all linear functionals T1, T2.
(b) Show that kT is a linear functional for all linear functionals T, and for all k ∊ 𝔽.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for the information sheet for Quiz 3.
Nov 14 Read your lecture notes and do
H56. Let T : ℝ3 → ℝ2 denote the linear map given by for all x, y, z ∊ ℝ.
Let {φ1, φ2, φ3} denote the dual basis of the standard basis of ℝ3 and let denote the dual basis of
the standard basis of ℝ2. Find and as linear combinations of φ1, φ2 and φ3.
H57. Let V denote the linear (vector) space of all polynomials in x, with coefficients in ℝ, and let T : V → V
denote the linear map given by for all x ∊ ℝ, where denotes the
2nd derivative of p with respect to x.
(a) Let φ ∊ be defined by where denotes the derivative of p with respect to x.
Find the linear functional on V.
(b) Let φ ∊ be defined by φ(p) = for all p ∊ V. Find
H58. Let V denote a linear (vector) space (possibly of infinite dimension).
(a) Show that the dual map of the zero map on V is the zero map on .
(b) Show that the dual map of the identity map on V is the identity map on
H59. Let V denote a linear (vector) space (possibly of infinite dimension). Suppose that T : V→ 𝔽 is a linear map.
If T is not the zero map, explain why T(V) = 𝔽.
H60. Suppose 𝔽 = ℝ and let V denote the linear space of all polynomials of degree at most five in the variable x
with coefficients in ℝ. Let U = {x2 f : f ∊ V, deg(f) ≤ 3} ∪ {0}.
(a) Verify that U is a subspace of V.
(b) Verify that x2 + U = x3 + U.

 

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Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for the information sheet for Quiz 3.
Nov 16 Read your lecture notes & the scan of the instructor’s lecture notes in onedrive (as some aspects were skipped in lecture owing to time constraints) and do
H60. (continued) Suppose 𝔽 = ℝ and let V denote the linear space of all polynomials of degree at most five in the
variable x with coefficients in ℝ. Let U = {x2 f : f ∊ V, deg(f) ≤ 3} ∪ {0}.
(c) Verify that 1 + U = x2 + 1 + U.
(d) Verify that xn + U = 0 + U for 2 ≤ n ≤ 5.
(e) Verify that x + U and 1 + U are distinct sets.
(f) Verify that x + U and x2 + U are distinct sets.
H61. (a) Suppose 𝔽 = ℝ and let V = ℝ2; find V/V.
(b) For any linear (vector) space V, find V/V.
H62. (a) Suppose 𝔽 = ℝ and let V = ℝ2; find V/{0}.
(b) For any linear (vector) space V, find V/{0}.
H63. Suppose 𝔽 = ℝ and let V = ℝ3 and
Verify that are linearly dependent in V/U .

H64. Suppose 𝔽 = ℝ and let V = ℝ4 and U =
(a) Verify that U is a subspace of V and find a basis of U.
(b) Find a basis B of V that contains the basis you found in (a).
(c) Write explicitly with respect to the basis B from (b).
(d) Find dim(V/U ).
(e) Find a basis of V/U and justify it is indeed a basis.
H65. Refer to question H60.
(a) Find dim(V).
(b) Find dim(U).
(c) Find dim(V/U).
(d) Given your answer in (c), find a basis of V/U.

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for the information sheet for Quiz 3.
Nov 21 Read your lecture notes & the scan of the instructor’s lecture notes in onedrive (as some aspects were skipped in lecture owing to time constraints), including page 66 in those lecture notes, and do
H66. Suppose 𝔽 = ℝ and let V = ℝ3, U = and T : V→V be given by A =
with respect to the standard basis of V.
(a) Verify that T(U) is contained in U.
(b) Find a basis of V/U and find dim(V/U).
(c) Find the matrix B of with respect to the basis you found in (b).
(d) Compare B with A; what do you notice?
H67. Suppose 𝔽 = ℝ and let V denote the linear space of all polynomials of degree at most three in the variable x
with coefficients in ℝ. Let U denote the subspace of V consisting of all the constant polynomials in V, and let
T = d/dx : V→V.
(a) Verify that T(U) is contained in U.
(b) Find the matrix A of T with respect to the basis {1, x, x2, x3} of V.
(c) Find the matrix B of the induced map with respect to the basis
{x + U, x2 + U, x3 + U} of V/U.
(d) Compare B with A; what do you notice?
H68. Suppose that T : ℝ x ℝ2 x ℝ3 → ℝ is a multilinear function (trilinear function) and that
T (1, e1, E1) = 2, T (1, e1, E2) = 4, T (1, e1, E3) = –1,
T (1, e2, E1) = –3, T (1, e2, E2) = 0, T (1, e2, E3) = 0,
where e1 = , e2 = , E1 = , E2 = , E3 = .
(a) Find . (b) Find .
H69. (a) Suppose that T : 𝔽 x 𝔽 x 𝔽 → 𝔽 is defined by T(a, b, c) = abc for all a, b, c ∊ 𝔽.
Show that T is a trilinear function.
(b) Let V1 , …, Vk , W be linear (vector) spaces, and suppose V1 = ··· = Vk . To say that a multilinear function
T : V1 x ··· x Vk → W is symmetric means that swapping any two distinct entries of the input preserves
the output; that is, T(v1, …, vi ,….vj ,…., vk) = T(v1, …, vj ,…. vi ,…., vk) for all vr ∊ V1 for all r,
where i < j. Show that the function T in (a) is symmetric.
H70. Let V1 , …, Vk be linear (vector) spaces and let Ti : Vi → 𝔽 denote linear maps for i = 1, …, k. Let
T : V1 x ··· x Vk → 𝔽 be defined by T(v1, …, vk) = T1(v1)T2(v2) ···Tk(vk) for all vr ∊ Vr for all r.
Show that T is a multilinear function.
H71. Let T : ℝ2 x ℝ2 → ℝ2 be defined by for all x1, x2, y1, y2 ∊ ℝ.
Verify that T is NOT a bilinear function.
H72. Let V1 denote the linear (vector) space of all polynomials in x with coefficients in ℝ and
let V2 denote the linear (vector) space of all polynomials in y with coefficients in ℝ and
let W denote the the linear (vector) space of all polynomials in the two variables x and y with coefficients
in ℝ.
(a) Show that the function T : V1 x V2 → W given by T(p, q) = pq , for all p ∊ V1, q ∊ V2, is a bilinear
function.
(b) For the function T in (a), show that x and y belong to the image of T, but x+y does not belong to the
image of T.
(c) Use (b) to show that the image of a multilinear function T : V1 x ··· x Vk → W need not be a subspace
of W if k ≥ 2.
H73. Let V1 , …, Vk , W be linear (vector) spaces, and suppose V1 = ··· = Vk . To say that a multilinear function
T : V1 x ··· x Vk → W is skew-symmetric means that swapping two distinct entries of the input multiplies the
output by –1; that is,
T(v1, …, vi ,….vj ,…., vk) = –T(v1, …, vj ,…. vi ,…., vk) for all vr ∊ V1 for all r, where i < j.
(Note that some authors call such a function alternating, but, strictly speaking, the definitions of
“skew-symmetric’’ and “alternating’’ can differ if .)
(a) Verify that if vi = vj in , then T(v1, …, vk) = 0 for all v1, …, vi-1 , vi+1 ,…, vj-1 , vj+1 ,…., vk ∊ V1.
(b) Show that, if W = 𝔽 and V1 = ··· = Vk = 𝔽k and if T is given by T(v1, …, vk) = det [v1 ··· vk] for all
v1, …, vk ∊ 𝔽k, then T is a skew-symmetric multilinear function.
(c) Let T : 𝔽2 x 𝔽2 →𝔽 be a skew-symmetric multilinear function (so k = 2, V1 = V2 = 𝔽2, W = 𝔽). Verify that
and .
(d) Continuing (c), write , and find .
(e) Continuing (c) and (d), show that for all a, b, c, d ∊ 𝔽.
Part (e) of the last question can be generalized as follows: if W = 𝔽 and V1 = ··· = Vk = 𝔽k and
T : V1 x ··· x Vk → 𝔽 is a skew-symmetric multilinear function, then there exists s ∊ 𝔽 such that
T(v1, …, vk) = s det [v1 ··· vk] for all v1, …, vk ∊ 𝔽k.

 

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 3; it is due Dec 5.

Nov 23

No lectures nor office hours today; UTA offices open until noon.
See the academic schedule at https://www.uta.edu/academics/academic-calendar
Nov 28 Read your lecture notes & the scan of the instructor’s lecture notes in onedrive (as some aspects were skipped in lecture owing to time constraints) and do
H74. Simplify in 𝔽4 ⊗𝔽 𝔽2.
H75. Simplify in 𝔽4 ⊗𝔽 𝔽2.
H76. Find in 𝔽2 ⊗𝔽 𝔽3.
H77. Find dim( 𝔽2 ⊗𝔽 𝔽7).
H78. The linear (vector) space 𝔽4 ⊗𝔽 𝔽 is isomorphic to a linear (vector) space of the form 𝔽n; find n.
H79. Expand 𝔽4 ⊗𝔽 (𝔽2 ⊕ 𝔽3).

 

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 3; it is due Dec 5.
Nov 30 Read your lecture notes & the scan of the instructor’s lecture notes in onedrive (as some aspects were skipped in lecture owing to time constraints) and do
H80. Let and .
Find .
H81. Let be given by .
Using the standard bases of 𝔽2 and 𝔽3 and their dual bases, find the element v ∊ 𝔽2 ⊗𝔽 𝔽3 to
which corresponds.
H82. Suppose that T1 is a linear map with eigenvalues 2, 4 and 5, and that T2 is a linear map with eigenvalues
3 and 6. Find the distinct eigenvalues of T1 ⊗ T2 .
H83. Suppose that T1 is a linear map with eigenspaces and , and
that T2 is a linear map with eigenspaces and .
Find the eigenspaces of T1 ⊗ T2 and the dimensions of those eigenspaces.
H84. In (a)-(h), find the Kronecker product A ⊗ B:
(a) A = and B = .
(b) A = and B = .
(c) A = and B = .
(d) A = and B = .
(e) A = and B = .
(f) A = and B = for all a, b, c, d ∊ 𝔽.
(g) A = and B = .
(h) A = and B = .
H85. Find the rank of the Kronecker product A ⊗ B for each of the following:
(a) A and B given by H84(a).
(b) A and B given by H84(b).
(c) A and B given by H84(e).
(d) A and B given by H84(g).
(e) A and B given by H84(h).
H86. Let v ∊ 𝔽n and w ∊ 𝔽m. Show that vt⊗ w = w vt, where ⊗ denotes the Kronecker product.
H87. Let A and B denote nxn matrices with entries in 𝔽 and let I denote the nxn identity matrix.
In (a)-(d), ⊗ denotes the Kronecker product.
(a) Show that (I ⊗ A)k = I ⊗ Ak and (B ⊗ I)k = Bk ⊗ I, for all k ∊ ℕ ∪ { 0 }.
(b) Use (a) to show that eI⊗A = I ⊗ eA and eB⊗I = eB ⊗ I.
(c) Show that the matrices I ⊗ A and B ⊗ I commute.
(d) Show that e(I ⊗ A)+(B ⊗ I ) = eB ⊗ eA.

 

If any of the mathematics does not display properly (e.g., an error message received), compare with this file.
Recall that a scan of the instructor’s lecture notes from Fall 2021 can be accessed via the Modules section of our Canvas portal.

Check Canvas for Quiz 3; it is due Dec 5.


Please remember to complete online the student feedback survey by 11 pm on Dec 6 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, lecture notes provided on onedrive, the Canvas graded homework (Q1 & Q2 & Q3), use of videos, corniness of jokes …??). Thank you!

I will have my usual office hours through Dec 5 inclusive, and I will also have additional office hours as noted below.
Dec 05 We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework.

 

Check Canvas for Quiz 3; it is due today.

Please remember to complete online the student feedback survey by 11 pm on Dec 6 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, lecture notes provided on onedrive, the Canvas graded homework (Q1 & Q2 & Q3), use of videos, corniness of jokes …??). Thank you!

The Final Test will be on Monday Dec 12; see Canvas for an information sheet.
I will have my usual office hours through today inclusive, and I will also have additional office hours as noted below.

Dec 07
1:15 PM

Office hour today 1:15-2:15 PM online only – see Canvas announcements for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.

Dec 09
1:20 PM

Office hour today 1:20-2:20 PM in PKH 462 and online – see Canvas announcements for the Teams link.
This is time the instructor is planning to be in PKH 462 and on Teams for students to “drop by” to ask questions.

Dec 10
2:45 PM

Office hour today 2:45-3:45 PM online only – see Canvas announcements for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.

Dec 12
2:00 PM

FINAL TEST today, starting at 2:00 PM; see Canvas for an information sheet.
Look over Tests 1-2 and Quizzes 1-3 and their solutions posted on Canvas to study for this test.
(Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)

The assignments from Fall 2021 can be viewed here .