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Linear Algebra
Fall 2015
Instructor: Kwangho Choiy
Office: Neckers 283 (618-453-6508)
E-mail: kchoiy_at_siu_dot_edu
Office Hours: Mon/Fri 12:00pm - 03:00pm, or by appointment. Emails are also available for simple questions.
Course Website: https://sites.google.com/site/kchoiy/home/teaching/previous-courses/math421
Class Meeting: MWF 10:00am - 10:50am in EGRA 420
Textbook: Linear Algebra 4th edition by S. H. Friedberg, A. J. Insel, and L. E. Spence.
Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule linked here [math421-syllabus and schedule], UPDATED on 8/20/2015.
Exams: There will be two mid-term exams and one final. No make-up exam will be accepted.
Exam 1 (in class on 9/28, Mon) covers Sec 1.2--1.6, 2.1
Exam 2 (in class on 10/28, Wed) covers Sec 2.2 -- 2.6.
Final Exam (10:15am-12:15pm on Dec 16, Wed) will be comprehensive: approximately, 40% on Sections 1.2-1.6, 2.1-2.6 and 60% on Sections 3.1-3.3, 4.1-4.3, 5.1, 5.2, 6.1, 6.2 ("6.3-6.6, 6.8" are excluded). (**See the sheet of practice problems of Final, which contains all details**)
Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date (see the Course Schedule 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.*
HW 1 (due on Sep. 4, Fri)
HW 2 (due on Sep. 16, Wed)
HW 3 (due on Sep. 23, Wed)
HW 4 (due on Oct. 9, Fri)
HW 5 (due on Oct. 16, Fri)
HW 6 (due on Oct. 23, Fri)
HW 7 (due on Nov. 9, Mon)
HW 8 (due on Nov. 18, Wed)
HW 9 (due on Dec. 2, Wed)
HW 10 (due on Dec. 9, Wed)
updates and remarks - math421 - Fall 2015:
[Dec 11, Fri] Lecture 43: Sec 6.2 -- orthogonal complement, V= direct sum of W and W per. A brief concepts of adjoint operator, normal operator from 6.3, a list of subjects of 6.3--6.6, 6.8. HW 10 returned with solution.
[Dec 9, Wed] Lecture 42: Sec 6.2 -- Gram-Schmidt orthogonality process, Fourier coefficients.
[Dec 7, Mon] Lecture 41: Sec 6.1 -- a proof of Cauchy-Schwarz inequality, more new terms and examples. The practice problems for Final distributed! -- read carefully.
[Dec 4, Fri] Lecture 40: Sec 6.1 -- examples of inner product space, motivation, norms, properties of inner product and norms. HW 9 returned with solution.
[Dec 2, Wed] Lecture 39: Sec 5.4 -- limits of a sequence consisting of matrices, some arguments of the convergence of limit of A^k with A, where A is n by n matrix whose entries are complex numbers, related to eigenvalues, a couple of examples // Sec 6.1 -- def of an inner product with F= R (real), C(complex). HW 10 (due on Dec. 9, Wed) distributed!
[Nov 30, Mon] Lecture 38: Sec 5.2 -- continued the proof of main theorem and two remarks / Sec 5.3 -- Cayley-Hamilton Theorem.
[Nov 23, Mon] Lecture 37: Sec 5.2 -- proof of main theorem, properties of eigenvectors.
[Nov 20, Fri] Lecture 36: Sec 5.2 -- main theorem (how to determine whether T is diagonalizable), some introduction to Chapter 6 and 7. HW 8 returned with solution. HW 9 (due on Dec. 2, Wed) distributed.
[Nov 18, Wed] Lecture 35: Sec 5.1 -- properties of the characteristic polynomial, the set of eigenvalues = the set of zeros of the characteristic polynomial, the null space of T-(eigenvalue)I contains an eigenvector corresponding to the eigenvalue, how to find all eigenvectors. Sec 5.2 -- def of multiplicity, eigenspace.
[Nov 16, Mon] Lecture 34: Sec 5.1 -- how to find eigenvectors and eigenvalues, introduce the characteristic polynomial.
[Nov 13, Fri] Lecture 33: Sec 5.1 -- motivation of diagonalization, introduced eigenvectors and eigenvalues for T and A. HW 7 returned with solution.
[Nov 9, Mon] Lecture 32: Sec 4.2 and 4.3 -- more properties of det, cramer's rule, its application to get the inverse A^{-1} of A, examples. HW 8 (due on Nov. 18, Wed) distributed!
[Nov 6, Fri] Lecture 31: Sec 3.3 -- the solution space of Ax=b is non-empty iff rank (A) = rank (A|b) / Sec 4.1 and 4.2 -- definition of determinant, some properties.
[Nov 4, Wed] Lecture 30: Sec 3.3 -- a few new terms regarding systems of linear equations, solution space K, "dim K for homogeneous Ax=0" = n-rank(A), "the solution space for non-homogeneous Ax=b" = a solution + "the solution space for the homogeneous Ax=0".
[Nov 2, Mon] Lecture 29: Sec 3.2 -- defined the rank of a matrix, rank(A) = dim (Span of column matrices of A) = dim (Span of row matrices of A), Ep....E2E1AG1G2...Gq = a matrix consisting of left-upper (the identity matrix I_rxr) and all zeros, with r = rank (A). HW 7 (due on Nov. 9, Mon) distributed!
[Oct 30, Fri] Lecture 28: Sec 3.1 -- elementary row (respectively, column) operations and elementary matrices, A'=EA (respectively, A'=AG). EXAM 2 returned with solution.
[Oct 28, Wed] Lecture 27: EXAM 2
[Oct 26, Mon] Lecture 26: Review for EXAM 2, HW6 returned with solution.
[Oct 23, Fri] Lecture 25: Sec 2.6 -- the transpose of T is connected to the transpose of [T], via the isomorphism between L(V,W) and in Mmxn(F), and example.
[Oct 21, Wed] Lecture 24: Sec 2.6 -- V ism to V**, corollary from this iso.The practice problems for EXAM 2 distributed!
[Oct 19, Mon] Lecture 23: Sec 2.6 -- linear functional, dual space V*, basis for V*, HW 5 returned with solution.
[Oct 16, Fri] Lecture 22: Sec 2.5 -- change of coordinates, A~B in Mnxn(F), A is similar to B. HW 6 (due on Oct. 23, Fri) distributed!
[Oct 14, Wed] Lecture 21: Sec 2.4 -- def of isomorphism of vector spaces, examples, proof of the theorem L(V,W) is isomorphic to Mmxn(F). HW4 returned (the solution will be distributed on Fri, Oct. 16).
[Oct 9, Fri] Lecture 20: Sec 2.4 -- invertible linear transformation, matrix, relation between two. HW 5 (due on Oct. 16, Fri) distributed!
[Oct 7, Fri] Lecture 19: Sec 2.3 -- continued to compare properties in L(V,W) and in Mmxn(F).
[Oct 5, Mon] Lecture 18: Sec 2.3 -- compare properties in L(V,W) and in Mmxn(F).
[Oct 2, Fri] Lecture 17: Sec 2.2 -- properties in L(V,W). HW 4 (due on Oct. 9, Fri) distributed!
[Sep 30, Wed] Lecture 16: Sec 2.2 -- ordered basis, coordinate vector, matrix representations of a linear transformation. EXAM 1 returned with solution.
[Sep 28, Mon] Lecture 15: EXAM 1
[Sep 25, Fri] Lecture 14: Review for EXAM 1, HW 3 returned with solution.
[Sep 23, Wed] Lecture 13: Sec 2.1 -- proved the dimension theorem, examples, more useful theorem involved in linear transformations.
[Sep 21, Mon] Lecture 12: Sec 2.1 -- more useful theorem involved in linear transformations, define null space, nullity, range, and rank, state the dimension theorem, the practice problems for EXAM 1 distributed!
[Sep 18, Fri] Lecture 11: Sec 2.1 -- motivation, def of linear transformation, examples, useful theorems, basic terms, HW2 returned with solution.
[Sep 16, Wed] Lecture 10: Sec 1.6 -- pf of the uniqueness, corollaries, dimension of subspaces. HW 3 (due on Sep. 23, Wed) distributed!
[Sep 14, Mon] Lecture 9: Sec 1.6 -- the uniqueness theorem for the number of vectors in finite base.
[Sep 11, Fri] Lecture 8: Sec 1.6 -- def of basis, examples, finite and infinite basis, two theorems regarding the basis.
[Sep 9, Wed] Lecture 7: Sec 1.5 -- how to extend a linearly indep subset, some new terms, useful techniques to show lin indep and dep, motivation for basis and dimension. HW1 returned with solution, HW2 (due on Sep. 16, Wed) distributed!
[Sep 4, Fri] Lecture 6: Sec 1.4 -- examples of Span(S), pf of Span(S) < V, / Sec 1.5 -- def of lin dep and lin ind, examples, some arguments related to Span(S) and lin dep
[Sep 2, Wed] Lecture 5: Sec 1.4 -- def of linear combinations, examples, system of equations, augmented matrices, Span(S) forms a subspace.
[Aug 31, Mon] Lecture 4: Sec 1.3 -- some interesting subspaces in M_nxn(F), sum, intersection, direct sum of subspaces / Sec 1.4 -- def of linear combinations.
[Aug 28, Fri] Lecture 3: Sec 1.3 -- definition of subspaces, examples, useful arguments for subspaces. HW 1 (due on Sep. 4, Fri) distributed!
[Aug 26, Wed] Lecture 2: Sec 1.2 -- definition of vector spaces, properties of vector spaces.
[Aug 24, Mon] Lecture 1: syllabus / Appendix C -- groups, fields / Sec 1.2 -- rough concept of vector spaces.
[Aug 23, Sun] Supplementary Material: Visit the link to see mathematical proofs, notation, arguments, etc.
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