Math421_S19

Linear Algebra

-Spring 2019-

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu

• Office Hours: T 9-10am; 11:15-12pm; 1:15-3pm / Th 9:15-10am; 11:15-12pm; 2-3pm, or by appointment. Emails are also available for simple questions.

Course Website: https://sites.google.com/site/kchoiy/teaching/math421_s19

Class Meeting: TTh 12:00pm - 1:15pm 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:  It is required to read carefully our syllabus and schedule linked [here], UPDATED on 1/10/2019.

Exams: There will be one mid-term and one final. No make-up exam will be accepted. 

1.  Mid-term (in class 12-1:15pm on 3/7, Thur) covers Sec 1.2-1.6, 2.1-2.4 (9 sections)

2.  Final (10:15am-12:15pm on May 9th, Thur) covers Sec 2.5-2.6, 3.1-3.3, 4.1-4.3, 5.1-5.2 (10 sections)  - announced in class and by email on 4/30

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.*

1. HW 1 (due on Jan. 24)

2. HW 2 (due on Jan. 31)

3. HW 3 (due on Feb. 7)

4. HW 4 (due on Feb. 14)

5. HW 5 (due on Feb. 21)

6. HW 6 (due on Feb. 28)

7. HW 7 (due on  Mar. 28)

8. HW 8 (due on Apr. 4)

9. HW 9 (due on Apr. 11)

10. HW 10 (due on Apr. 18)

11. HW 11 (no need to submit)

Updates/Remarks - Math421 Spring 2019:

[May 2, Thu] Sec 5.1-5.4 - proof of the main theorem for diagonalization, examples / Sec 6.1-6.8 - notes emailed out.

[Apr 30, Tue] Sec 5.1-5.4 - continue with how to find the invertible matrix Q, how to choose two ordered bases, examples, eigenvalues, eigenvectors, nullity(A-\alpha_i) =< m_i. 

[Apr 25, Thu] Sec 5.1-5.4 - eigenspace, examples, how to obtain the diagonal matrix, how to find the invertible matrix Q using relations between [T]_\beta and [T]_\beta'.

[Apr 23, Tue] Sec 5.1-5.4 - diagonalizable, non-diagonalizable examples, characteristic polynomial, remarks on zeros and fields related to char. ploy. Graded HW 10 with solution is returned.

[Apr 18, Thu] Sec 4.1-4.3 - Cramer's rule, example / Sec 5.1-5.2 - motivations, big picture, definition of diagonalizable A, T, examples, set up the objective, gave a technique without proof, example. HW 10 (no need to submit) is distributed.

[Apr 16, Tue] Sec 4.1-4.3 - motive, formal definition of det(A), examples, properties of det(A), give an explicit solution to Ax=b using det(A). Graded HW 9 & Quiz 9 with solutions are returned.

[Apr 11, Thu] Sec 3.3- 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, consistent, inconsistent, examples. HW 10 (due on Apr 18) is distributed / Quiz 9 is taken.

[Apr 9, Tue] Sec 3.3 - new terms regarding systems of linear equations, solution space(set) K, the solution set for non-homogeneous Ax=b is of the form, a solution + the solution space for the homogeneous Ax=0. Graded HW 8 & Quiz 8 with solutions are returned.

[Apr 4, Thu] Sec 3.2- further properties related to rank, multiple of elementary matrices left(row) right(column) towards a nice form with rank,  example. HW 9 (due on Apr 11) is distributed / Quiz 8 is taken.

[Apr 2, Tue] Sec 3.2 - L_A, construct T such that [T]=A, rank(A)=rank(L_A), examples, properties related to rank. Graded HW 7 & Quiz 7 with solutions are returned with comments.

[Mar 28, Thu] Sec 2.6- proof of [^tT] = [T]^t, example / Sec 3.1 - for a general field F, elementary operations type 1,2,3, elementary matrices, example. HW 8 (due on Apr 4) is distributed / Quiz 7 is taken.

[Mar 26, Tue] Sec 2.6 -  introduced V**, double dual space, statements and proofs about bases of V* and V**, transpose of T, relation between L(V,W) and L(W*,V*).

[Mar 21, Thu] Sec 2.5 - example for relations between [T]_\beta and [T]_\beta'; [x]_\beta and [x]_\beta', introduced similar matrices / Sec 2.6 - introduced V*, dual space, linear functional, defined and found basis for V*. HW 7 (due on  Mar. 28) is distributed.

[Mar. 19, Tue] Reviewed Mid-term / Sec 2.5 - stated, proved the relation between [T]_\beta and [T]_\beta', example. Returned graded mid0terms with solution.

[Mar 7, Thu] Mid-term taken

[Mar 5, Tue] Sec 2.5 - relations between [T]_\beta and [T]_\beta'; [x]_\beta and [x]_\beta' / Review for Mid-term - summarized topics in the coverage. Graded HW 6 & Quiz 6 with solutions are returned with comments.

[Feb. 28, Thu] Sec 2.4 - several arguments related to invertible transformations and isomorphism. Quiz 6 is taken / mid-term announced in class and by E-mail.

[Feb 26, Tue] Sec 2.4 - more interpretation and examples for isomorphism, techniques to verify an isomorphism, definition and motivation of invertible linear transformation, remarks for invertible linear transformation. Graded HW 5 & Quiz 5 with solutions are returned with comments.

[Feb. 21, Thu] Sec 2.3 - review the key argument for [T]_\beta^\gamma, proof of [TU]_\beta^\delta=[T]_\gamma^\delta[U]_\beta^\gamma, example / Sec 2.4 - definition of vector space isomorphism, example, L(V), [T]_\beta. HW 6 (due on Feb. 28) is distributed / Quiz 5 is taken.

[Feb 19, Tue] Sec 2.3 - structure in the vector space L(V,W), zero transformation T_0, which is the zero vector in  L(V,W), introduced isomorphism, L(V,W) is isomorophic to M_mxn(F), three natural questions related to L(V,W) based on structure of M_mxn(F). Graded HW 4 & Quiz 4 with solutions are returned.

[Feb. 14, Thu] Sec 2.2 - ordered basis, recall the unique existence of coefficients, [x]_\beta, [T]_\beta^\gamma, examples, L(V,W), the connection between L(V,W) and Mat_mxn(F). HW 5 (due on Feb. 21) is distributed / Quiz 4 is taken.

[Feb 12, Tue] Sec 2.1 - proof of dim V = nullity(T) + rank(T), example, remarks including T(0)=0 iff T is injective / Sec 2.2 - motivation, example of matrix representations of linear transformations. Graded HW 3 & Quiz 3 with solutions are returned with some comments.

[Feb. 7, Thu] Sec 2.1 - more examples for linear transformations, properties of linear transformations including two criteria equivalent to being linear, T(0)=0, T(subspace) is subspace, T^{-1}(subspace) is subspace, R(T), the range of T, N(T), rank, the null space of T, nullity, dim V = dim N(T) + dim R(T). HW 4 (due on Feb. 14) is distributed / Quiz 3 is taken.

[Feb 5, Tue] Sec 1.6 - proof of the equality of two base in cardinality, remark / Sec 2.1 - definition of linear transformations, remark. Graded HW 2 & Quiz 2 with solutions are returned with some comments.

[Jan. 31, Thu] Sec 1.6 - more examples for linear independent, def of a basis, examples, arguments related to base. HW 2 (due on Feb. 7) is distributed / Quiz 2 is taken.

[Jan 29, Tue] Sec 1.5 - linearly independent, dependent, examples, properties of liner independence. Graded HW 1 & Quiz 1 with solutions are returned with some comments.

[Jan. 24, Thu] Sec 1.2 - more examples for subspaces, two equivalent criteria to check subgroups, W_1 \cap W_2, W_1+W_2 / Sec 1.4 - linear combination, Span(X) < V, examples, proofs. HW 2 (due on Jan. 31) is distributed / Quiz 1 is taken.

[Jan. 22, Tue] Sec 1.2 - some further proofs and remarks of properties of v.s. including cancellation law / Sec 1.3 - def of subspace, example, remarks on the list of v.s. axioms for the subspace's sake, 3 statements to be checked for being a subspace.

[Jan. 17, Thu] Sec 1.2 - structure of fields, several examples of v.s/F, properties of v.s/F, sketch of proof. HW 1 (due on Jan. 24) is handed out.

[Jan. 15, Tue] syllabus distributed; Sec 1.2 - 3 motivated examples from vector spaces over R, recalled definition and notes for fields, definition of vector spaces over a field with axioms,