Book =

Linear Algebra (Pure & Applied Undergraduate Texts) by Michael E. Taylor, American Math. Soc., 2020.
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Aug 25

Check your Canvas notifications to check you can receive Canvas announcements.
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 Canvas course website.
Read https://vancliff.utasites.cloud/T/study.pdf 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 can be found in the Modules section of our Canvas portal.

Aug 30

Read your lecture notes, skim/read Section 2.1 and do Sec 2.1: 11, 8, 9.
Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Sep 01

Read your lecture notes and do Sec 2.2: 1, 2.
Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Sep 06

Labor Day Holiday (no lecture)

Sep 08

Read your lecture notes and do
H1. Repeat Questions 2.2.1 and 2.2.2 for the matrix A₄ , where

Sec 2.2: 3.

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Sep 13

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 A₅^k = 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 A_i^k = 0.
Sec 2.3: 2, 3 (assume the notation in question 3 is referring to question 2).

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Sep 15

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 A^k = 0 iff B^k = 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 can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 1 ; it is due Sept 22.

Sep 20

Read your lecture notes. You might also want to read the file on JCF used in Math 5333 (optional).
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.

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 can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 1; it is due Sept 22.

Sep 22

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 ).

Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 1; it is due today.

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

Sep 27

Read your lecture notes and do
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?
H19. Assume, without proof, that the pairing   for all   a_i, b_i ∊ 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 e₁ = ∊ V and e₂ = (4x – 3) ∊ V.
          (a) Justify that e₁ and e₂ are orthonormal.
          (b) Use the Gram-Schmidt formula (GSOP) to find 0 ≠ e₃ ∊ V such that e₁, e₂ and e₃ 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 can be found in the Modules section of our Canvas portal.

Test 1 will be on Wednesday; check Canvas for an information sheet.

Sep 29

Test 1 today.

Oct 04

Read your lecture notes and look over the solutions to Test 1 in the Assignments section of our Canvas portal, and do
H21. Suppose 𝔽 = ℂ and let T : ℂ³ → ℂ² be defined by   
          where i² = –1.
         (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 v_i ∊ 𝔽 for all i and v_i = 0 for all but finitely many i.
          Let e_i = (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*(e_n) = (z₁, z₂, ....) where z_i ∊ 𝔽 for all i, but z_i is nonzero for infinitely many i (so T*(e_n) ∉ 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 can be found in the Modules section of our Canvas portal.

Oct 06

Read your lecture notes and do
H28. Show that   A^tA-AA^t   is a symmetric matrix for all nxn matrices A (recall A^t 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) and
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 can be found in the Modules section of our Canvas portal.

Oct 11

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.

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Oct 13

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.

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Oct 18

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.
       (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).

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 can be found in the Modules section of our Canvas portal.

Oct 20

Read your lecture notes, especially the examples of polar decomposition (PD) and singular-value decomposition (SVD), and read the example in the book that appears right after Proposition 3.6.7.
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.
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 = UDV^t, 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 = UDV^t, 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 UV^t = 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 can be found in the Modules section of our Canvas portal.

Test 2 will be 1 week from today; check Canvas for an information sheet late in the day on Oct 20 or on Oct 21.

Oct 25

Read your lecture notes and do
H47. Find e^It where I is the 2x2 identity matrix.

Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Test 2 will be on Wednesday; check Canvas for an information sheet late in the day on Oct 20 or on Oct 21.

Oct 27

Test 2 today.

Nov 01

Read your lecture notes and look over the solutions to Test 2 in the Assignments section of our Canvas portal, and do
Sec 3.7: 16.

Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Nov 03

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
        

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Nov 08

Read your lecture notes and do
H49. Suppose 𝔽 = ℝ. In ℝ⁴, let    Find u ∊ U such that   is as small as possible.

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.

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Recall that a scan of the instructor's lecture notes can be found in the Modules section of our Canvas portal.

Nov 10

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 2x2 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 2x2 matrices with entries in ℝ, T(M) = the 22-entry of M, for all M ∊ V.
         (e) V = the space of all 2x2 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   (ℝ³)'  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 , p⁽¹⁾ = p' , p⁽²⁾ = p'', etc, and 0! = 1.
         Show that  { }  is the dual basis of V' 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 V' relative to the basis in (a).

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 can be found in the Modules section of our Canvas portal.

Nov 15

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  {ψ₁ , ψ₂}  denote the dual basis of
         the standard basis of ℝ².    Find  T'(ψ₁)  and  T'(ψ₂)  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  (T(p))(x) = x p(x) + 2 p''(x)  for all x ∊ ℝ, where p'' denotes the 2^nd derivative of p with respect to x.
         (a) Let φ ∊ V' be defined by  φ(p) = p'(3),  where p' denotes the derivative of p with respect to x.
               Find the linear functional T'(φ) on V.
         (b) Let φ ∊ V' be defined by  φ(p) =  for all p ∊ V.  Find   (T'(φ))(x²).
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 V'.
        (b) Show that the dual map of the identity map on V is the identity map on V'.
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.
         (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.

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 can be found in the Modules section of our Canvas portal.

Check Canvas for an information sheet for Quiz 2.

Nov 17

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
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.
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?

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 can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 2; it is due Nov 29.

Nov 22

Read your lecture notes and do
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 , …, V_k , W  be linear (vector) spaces, and suppose V₁ = ··· = V_k . To say that a multilinear function   
              T : V₁ x ··· x V_k → W   is symmetric means that swapping any two distinct entries of the input preserves the output;
              that is, T(v₁, …, v_i ,….v_j ,…., v_k) = T(v₁, …, v_j ,…. v_i ,…., v_k)   for all v_r ∊ V₁ for all r, where i < j.
              Show that the function T in (a) is symmetric.
H70. Let V₁ , …, V_k be linear (vector) spaces and let  T_i : V_i → 𝔽  denote linear maps for i = 1, …, k. Let
         T : V₁ x ··· x V_k → 𝔽  be defined by    T(v₁, …, v_k) = T₁(v₁)T₂(v₂) ···T_k(v_k)   for all v_r ∊ V_r 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 V_k → W  need not be a subspace of W if k ≥ 2.
H73. Let V₁ , …, V_k , W  be linear (vector) spaces, and suppose V₁ = ··· = V_k . To say that a multilinear function  
         T : V₁ x ··· x V_k → W is skew-symmetric means that swapping two distinct entries of the input multiplies the output by –1;
         that is, T(v₁, …, v_i ,….v_j ,…., v_k) = –T(v₁, …, v_j ,…. v_i ,…., v_k)   for all v_r ∊ 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  v_i = v_j  in , then  T(v₁, …, v_k) = 0  for all v₁, …, v_i-1 , v_i+1 ,…, v_j-1 , v_j+1 ,…., v_k ∊ V₁.
        (b) Show that, if W = 𝔽 and V₁ = ··· = V_k = 𝔽^k and if T is given by  T(v₁, …, v_k) = det [v₁ ··· v_k]   for all v₁, …, v_k ∊ 𝔽^k,
              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 = ··· = V_k = 𝔽^k   and  T : V₁ x ··· x V_k → 𝔽  is a
skew-symmetric multilinear function, then there exists s ∊ 𝔽 such that  T(v₁, …, v_k) = s det [v₁ ··· v_k]   for all  v₁, …, v_k ∊ 𝔽^k.

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 can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 2; it is due Nov 29.

Nov 24

No lectures or office hours today; UTA offices open.
See the academic schedule at https://www.uta.edu/academics/academic-calendar/fall-2021

Nov 29

Read your lecture notes 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 can be found in the Modules section of our Canvas portal.

Check Canvas for Quiz 2; it is due Nov 29.

Dec 01

Read your lecture notes and watch the 28-minute video on the Kronecker product (use the link provided or access it in the Modules section of our Canvas portal) and then 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 ∊ 𝔽^m. Show that  v^t⊗ w = w v^t, 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 ⊗ A^k   and   (B ⊗ I)^k = B^k ⊗ I,  for all k ∊ ℕ ∪ { 0 }.
          (b)  Use (a) to show that  e^I⊗A = I ⊗ e^A  and   e^B⊗I = e^B ⊗ I.
          (c)  Show that the matrices  I ⊗ A  and  B ⊗ I  commute.
          (d)  Show that   e^{(I ⊗ A)+(B ⊗ I )} = e^B ⊗ e^A.