s20-math421

Linear Algebra (math 421)

-Spring 2020-

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu

• Office Hours: MW10-12; M1-2;F11-12, or by appointment. Emails are also available for simple questions.

Course Website: https://sites.google.com/site/kchoiy/teaching/s20-math421

Class Meeting: MWF 12:00pm - 12:50pm in EGRA 222

Textbook/Topics: Linear Algebra 4th edition by S. H. Friedberg, A. J. Insel, and L. E. Spence. The topics include vector spaces, linear independence, bases, linear transformations and matrices, eigenvalues and eigenvectors, inner product spaces, orthogonality, the Gram-Schmidt process, adjoint operators, Jordan canonical forms, quadratic forms, Sylvesters law of inertia - Sections 1.2–1.6; 2.1–2.6; 3.1–3.4; 4.1–4.4; 5.1, 5.2, 5.4; 6.1–6.6, 6.8; 7.1, 7.2.

Syllabus / Course Schedule: Available at D2L.

Exams: There will be one mid-term and one final. No make-up exam will be accepted. The following coverage and schedule may be subject to change, but they will be confirmed at least one week ahead of time:  

1.  Mid-term (in class, Mar. 6th, FRI) covers Sections 1.2 – 1.6, 2.1–2.6 - practice problems distributed in class on 2/28 // result informed by E-mail individually + solution uploaded at D2L. 

2.  Final Exam consists of Part A and Part B. *emailed/uploaded with detailed guidance at D2L 4/21 // result with grading + solution posted/uploaded at D2L after the final in the week of 5/4.

Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date (see the Course Schedule linked above for a tentative assignment schedule!):

*HW Policies: You should show all your work and submit it in class on the due date. No late homework will be accepted.*

1. HW 1 (due on 1/24, Fri) - handed out/uploaded at D2L 

2. HW 2 (due on 1/31, Fri) - handed out/uploaded at D2L 

3. HW 3 (due on 2/7, Fri) - handed out/uploaded at D2L 

4. HW 4 (due on 2/14, Fri) - handed out/uploaded at D2L 

5. HW 5 (due on 2/21, Fri) - handed out/uploaded at D2L 

6. HW 6 (due on 2/28, Fri) - handed out/uploaded at D2L 

7. HW 8 (due on 4/3, Fri) - assigned/uploaded at D2L, 3/25, with detailed guidance for online submission via D2L. 

8. HW 9 (due on 4/10, Fri) - assigned/uploaded at D2L, 3/31, with detailed guidance for online submission via D2L. 

9. HW 10 (due on 4/17, Fri) - assigned/uploaded at D2L, 4/7, with detailed guidance for online submission via D2L.

10. HW 11 (due on 4/24, Fri) - assigned/uploaded at D2L, 4/15, with detailed guidance for online submission via D2L.

Updates and Remarks - MATH421 - Spring 2020:

[5.1] Sec 6.1-6.2 motives for orthonormal basis, outline of strategy of creating orthonormal basis from a given basis for finite, infinite-dimemsional v.s., arguments related orthogonal subset consisting of non-zero vectors, Gram-Schmidt process, examples, closing remarks. (*The lecturenotes video includes some other parts of Sec 6) / Lecture videos available at D2L 4/25

[4.29] Sec 6.1-6.2 properties and proofs of <,> and || including linear properties on the 1st and 2nd component of <,>, Cauchy-Schwarz inequality, triangle inequality, examples, orthogonal, orthonormal, normalizing, remark about how to make an orthonormal ordered basis from a given arbitrary basis for F-inner product space. Lecture videos available at D2L 4/25

[4.27] Sec 6.1-6.2 further examples for <,> and proofs, F-inner product space, norm, properties of <,>. Lecture videos available at D2L 4/25

[4.24] Sec 6.1-6.2 motivating example of dot product in V=R^2, F=R, length, angle from the dot product, objective, general definition of the inner product on a general v.s. V over F=R,C as a function VxV->F with 4 axioms, examples, verifications. Lecture videos available at D2L 4/23

[4.22] Sec 5.3-5.4 sketch of proofs, further comments for Caley-Hamilton, limit of sequence of matrices, existence of A^k using eigenvalue of A as an application / Sec 6-7 outline of topics, motivations of Sec 7.1-7.4, rational canonical forms, Jordan canonical forms, minimal polynomial, and Sec 6.1-6.8, inner product, inner product space, etc. Lecture videos available at D2L 4/21

[4.21] Final Exam consists of Part A and Part B. *emailed/uploaded with detailed guidance at D2L.

[4.20] Sec 5.2 further comments for the criterion, examples for finding beta, [T]_beta, Q / Sec 5.3-5.4 Caley-Hamilton, examples, limit of sequence of matrices, existence of A^k using eigenvalue of A as an application. Lecture videos available at D2L 4/19

[4.17] Sec 5.2 multiplicity, split, examples, distinct dimV eigenvalues=>diagonalizable, 1 =< dim E_lambda =< multi of lambda, criterion determining whether T is diagonalizable, proofs. Lecture videos available at D2L 4/17

[4.15] Sec 5.2 summarize two requirements/expectations towards diagonalization, stated/proved theorems related to properties of eigenvectors, eigenvalues, characteristic polynomials, three relevant corollary/remarks. Lecture videos available at D2L 4/14

[4.13] Sec 5.2 motive to study factorization of f_T(t), f_A(t) into linear terms with multiplicity of each zero, and each ordered basis of each eigenspace given distinct eigenvalues, motivating example/statement, stated/proved independence of eigenvectors with distinct eigenvalues. Lecture videos available at D2L 4/12

[4.11] Catching-Up during the last one week (4/6-4/10) materials emailed/uploaded at D2L

[4.10] Sec 5.1-5.2 set up two goals, how to get eigenvectors from eigenvalues properties, N(T-lambda I_V), characterization of eigenvectors, examples, remakrs 1),1’),1’’), 2) in eigenvectors, eigenspace, nullspace, nullity, basis elements in each eigenspace, diagonalization. Lecture videos available at D2L 4/9

[4.8] Sec 5.1 properties of characteristic polynomial f_A(t) and f_T(t) in terms of A and T, respectively, relation between roots of char. poly and eigenvalues, independence of choices of ordered basis alpha in f_T(t), examples including two examples of # of eigenvalues =0, n, depending of F. Lecture videos available at D2L 4/7

[4.6] Sec 5.1 examples for eigenvector/value/diagonalizable in terms of both T and A, how to find eigenvector/value, motivating example for characteristic polynomial, defined characteristic polynomial in terms of A and T, examples. Lecture videos available at D2L 4/5

[4.3] Catching-Up during the last two weeks (3/23-4/3) materials emailed/uploaded at D2L

[4.3] Sec 5.1 why eigenvectors and values is to give a diagonalization with finite-dimensional v.s. V in terms of [T]_beta, if beta can be consisting of eigenvectors, three remarks from this experience, definition of diagonalizable T with dim V finite, definition of eigenvector/value/diagonalizable in terms of A in M_n(F), setting T=L_A, coincidence with 221 knowledge of existing of Q such that Q^{-1}AQ is a diagonal matrix by taking Q=[I_{F^n}]_beta^{std}, where beta consists of eigenvectors. Lecture videos available at D2L 4/2

[4.1] Sec 5.1 motivation for diagonalization, several relevant natural questions, narrow down to a single question - finding a ‘nice’ ordered basis beta to make [T]_beta to be a diagonal matrix, a table describing differences between 221 and 421 regarding diaognalization full general definition of det of nxn matrix, definition of eigenvalues/vectors for T in L(V) including infinite-dimensional v.s. V, two further questions how those defs related to the main motive and how to be defined in terms of A in M_n(F). Lecture videos available at D2L 3/31

[3.30] Sec 4.1-4.3 - motive, full general definition of det of nxn matrix, recall 221 case-by-case definition of det(A), new term, the full general definition of det(A), examples, several properties of det(A) and their proofs, give an explicit solution to Ax=b using det(A), application to solving Ax=b (Cramer's rule), example. Lecture videos available at D2L 3/29

[3.27] Sec 3.3 Ax=b, consistent, inconsistent, examples, homogeneous, non-homogeneous, solution set, solution space, K_H = N(L_A), dim K_H = n - rk(A), K is non-empty or empty iff respectively rk(A)=rk(A|b) or not, several useful arguments related to Ax=b, examples. Lecture videos available at D2L 3/26

[3.25] Sec 3.2 details about rank(A)=rank(L_A)=dim(column space)=dim(row space), Theorem 3.6, further properties related to rank, multiple of elementary matrices left(row) right(column) towards a nice form with rank, example. Lecture videos available at D2L 3/24

[3.23] Sec 3.1 L_A for any given A in Mat_{mxn}(F), so that [L_A]=A w.r.t std ordered base for V=F^n, W=F^m and L_A in L(F^n,F^m), which indicates the backward of the isomorphism between L(V,W) and Mat_mxn(F), rank(A)=rank(L_A)=dim(column space)=dim(row space), examples. Lecture videos available at D2L 3/20 / Online 421 guideline after the spring break sent 3/19.

[3.6] midterm Exam taken.

[3.4] Review Session for midterm Exam. Summarized all topics, Emailed partial solutions to the prac. Probl.

[3.2] Sec 2.6 based on examples for V*, constructed a basis beta*={f_i} for V* with dimV=n, ordered basis beta={x_i} such that f_i(x_j)=delta_ij, gave a canonical isomorphism V and V**, [T*]_gamma*^beta*=([T]_beta^gamma)^t, where T*(g)=gT, T in L(V,W) T* in L(W*,V*), ordered base beta for V, gamma for W, respectively.

[2.28] Sec 2.6 examples for V*, basis for V* with dimV=n, three main objectives. Quiz 6 is taken / Mid-term exam announced in class/by email/at D2L with practice problems handed-out.

[2.26] Sec 2.6 definition of V*, V**, dual space, double dual space, V~V**, relations between V, V*, V**. The full summary of Chapter 1 and 2 is given / Graded HW 5, Quiz 5 returned / HW 5 Solution is emailed.

[2.24] Sec 2.5 - [T]_beta^gamma =[I_W]_gamma'^gamma[T]_beta'^gamma'[I_V]_beta^beta' / Sec 2.6 - introduced a dual space.

[2.21] Sec 2.5 [x]_beta=[I_V]_beta’^beta [x]_beta’, example, technique to evaluate [I_V], motive, definition of invertible transformation, [T^{-1}]_beta^gamma =  [T]_beta^gamma^{-1}. Graded HW 4, Quiz 4 returned / Quiz 5 taken / HW 6 handed out/uploaded at D2L.

[2.19] Sec 2.5 examples of coordinate matrix, matrix representation with different ordered base, motive of changes of coordinate matrices, [x]_beta=[I_V]_beta’^beta [x]_beta’, example.

[2.17] Sec 2.3-2.4 motive of isomorphism, defined isomorphism of v.s., proved L(V,W) is iso to Mat_mxn(F), corollary.

[2.14] Sec 2.2-2.3 introduced L(V,W), defined the vector addition T+U and scalar multiplication rT, motivated from matrix addition and scalar multiplication in Mat_nxm(F) via the matrix representation, [T+rU]_beta^gamma=[T]_beta^gamma+r[U]_beta^gamma. Graded Quiz 3 returned / Quiz 4 taken / HW 5 handed out/uploaded at D2L.

[2.12] Sec 2.1 - another proof of dimension thm / Sec 2.2-2.3 ordered basis, coordinate matrix, matrix representation, examples. Graded HW 3 returned with solution.

[2.10] Sec 2.1 - proof of dimension theorem, two more properties, proofs related to T, examples.

[2.7] Sec 2.1 - motivation, definition of linear transformations, examples, N(T), R(T), nullity, rank, dimension theorem. Quiz 3 taken / HW 4 handed out/uploaded at D2L.

[2.5] Sec 1.6 - more properties and their proofs related to a basis. Graded Quiz 2 and HW 2 returned with solution.

[2.3] Sec 1.6 - more examples for base, listed/proved properties related to a basis

[1.31] Sec 1.6 - defined a basis, examples, |B_1|=|B_2|.  Quiz 2 taken / HW 3 handed out/uploaded at D2L.

[1.29] Sec 1.4-1.5 - more examples for Span, linearly independent, conventions, properties.  Graded Quiz 1 and HW 1 returned with solution.

[1.27] Sec 1.3 - proof of subspace criterion, more remarks on subspace criterion / Sec 1.4-1.5 - defined Span(X), linearly independent, examples for Span(X), Span(X) < V.

[1.24] Sec 1.3 - more examples for subspaces, properties of subspaces: another shorter subspace criterion, W_1 cap W_2, W_1+W_2, examples.  Quiz 1 taken / HW 2 handed out/uploaded at D2L.

[1.22] Sec 1.2 - proof of properties of vector spaces / Sec. 1.3 - defined subspace, gave subspace criterion, example of subspace.

[1.17] Sec 1.2 - examples for v/s, proofs, properties of v.s.  HW 1 handed out/uploaded at D2L.

[1.15] Sec 1.2 - more about fields, examples, definition of a vector space over a field, 8 axioms, example, checking VS1, 3 with R^2, zero vector.

[1.13] Course syllabus/schedule/objectives are given / Sec 1.2 - recalled vector spaces over R, Appendix C fields, axioms, examples. Syllabus/schedule handed out/emailed/uploaded at D2L.