LECTURE: MATH 3A, FALL 2016, M-W-F: 2-2:50 @ RH 114
INSTRUCTOR: NAM TRANG
OFFICE HOURS: M 3--4, W 1:15--2 & 3--3:45 @ RH 410N
OFFICE HOURS (during Final's week): W:3-6, F:11-12 @ RH 410N

DISCUSSION: T-Th: 2-2:50 @ RH 114
TA: D.A. BERGMAN
OFFICE HOURS: M 4-5, Th 4-5 @ RH 540M


COURSE POLICIES     COURSE SYLLABUS     UCI Academic Honesty Policies     UCI Student Resources

HOMEWORK ASSIGNMENTS:     HW1    HW2     HW3     HW4 (now due Nov 9)     HW5     HW6

MIDTERM/FINAL REVIEWS AND SAMPLE FINAL:     Midterm review    Final review     Sample final

FINAL'S WEEK:

QUIZZES:

Quiz 1: Thursday Sep 29. Sections: 1.1 and 1.2.
Quiz 2: Thursday Oct 6. Sections: 1.3 and 1.4.
Quiz 3: Thursday Oct 13. Sections: 1.5 and 1.7.
Quiz 4: Thursday Oct 20. Sections: 1.8 and 1.9.
Quiz 5: Thursday Oct 27. Sections: 2.1 and 2.2.
Quiz 6: Thursday Nov 9. Sections: 2.8 and 2.9.
Quiz 7: Thursday Nov 16. Sections: 3.1 and 3.2.
Quiz 8: Tuesday Nov 22. Sections: 3.3 and 5.1.
Quiz 9: Thursday Dec 1. Sections: 5.2 and 5.3.

COURSE PROGRESS

Week 0
F: Went over the syllabus, course policies, and goals for the course; defined systems of linear equations and gave various examples of geometric interpretations of solving a system of linear equations (i.e. finding intersections of lines, planes ...).

Week 1
M: Introduced matrices, elementary row operations, row-echelon forms/row-reduced echelon forms, pivots, and went through an example of solving a linear system of equations by row reducing the corresponding augmented matrix to its row echelon form.
W: Went through various examples of solving systems of linear equations: the cases with exactly 1 solution, with infinitely many solutions, and with no solutions. Basically finished Section 1.2.
F: Introduced vectors, defined vector addition and scalar multiplication, defined span, and went over a problem of deciding whether a vector w is in the span of v_1, ... , v_k.

Week 2
M: Introduce matrix equation: A x= b where A is mxn matrix, x is a vector in R^n, b is a vector in R^m; show how to multiply an mxn matrix A with a vector x in R^n; show how to convert a system of linear equations to a vector equation and to a matrix equation; introduce propositional logic and truth tables of A or B, A and B, negation of A, if A then B, A if and only if B.
W: Continue with truth tables/proof methods, proved theorem 4 in Section 1.4.
F: Returned homework, described how to represent the solution set of Ax=b in terms of the solution set of Ax=0 (so every solution of Ax=b is for the form x = v + x_h where v is a particular solution to Ax=b and x_h is a solution to Ax=0); defined linear dependence and linear indepdendence.

Week 3
M: Continue with linear dependence/linear independence, gave examples of when some vectors are linearly independent/linearly dependent, showed that {v} is linearly independent iff v is not the 0-vector; in the middle of showing {v_1,v_2} is linearly independent iff one vector is a scalar multiple of the other.
W: Finished linear independence/dependence section; showed that v_1...v_k are linearly dependent if they are vectors in R^n and k>n; showed that if {v_1...v_k} contains the zero-vector, then they are linearly dependent; went through an example of computing equilibrium solutions (in Section 1.6).
F: Reviewed functions, one-to-one, onto functions; defined linear transformation; gave 1 example of how to show if some function is NOT a linear transformation (one needs to give an explicit counter-example); gave 1 example of how to show if a function is a linear transformation (the function T_A(x) = A.x is a linear transformation for some fixed matrix A).

Week 4
M: Continue with linear transformations, gave examples of linear transformations: projections, dilations, rotations, proved the theorem that states: if T:R^n-->R^m is a linear transformation, then there is a mxn matrix A such that T(x) = Ax for any vector x in R^n.
W: Continue with linear transformations, gave complete argument that the projection to the x-axis is a linear transformation and computed its matrix, gave characterizations: T:R^n--> R^m is onto if the column vectors of the matrix A_t span R^m, T is one-to-one iff the columns of A_T are linearly independent.
F: Operations on matrices (addition, multiplication, transpose, power).

Week 5
M: Defined inverse of a matrix and gave an algorithm for determining if a matrix has an inverse and computing its inverse.
W: Gave examples of applying the algorithm for determining whether a matrix A is invertible; went through the invertible matrix theorem.
F: Invertible matrix theorem (cont.); explained why inverse of a matrix A is designed to model after the inverse function of the map T_A.

Week 6
M: Midterm
W: Defined column space, row space, and null space. Defined basis of a subspace.
F: Gave examples of how to determine if a set of vectors forms a basis; gave an algorithm for computing the column space of a matrix A.

Week 7
M: Finished 2.9 (defined dimension, rank, defined coordinates of a vector relative to a basis and gave examples).
W: Defined determinant of a matrix, stated Lagrange theorem (expansion along any row or any column gives the same result), stated and explained consequences (determinant of a triangular matrix is the product of the diagonal entries, determinant of A^T is determinant of A), examined how performing row operations on A changes the determinant of A.
F: Veteran's Day. No class.

Week 8
M: Described an algorithm for computing det(A) (row reduce A to a REF U of A without multiplying any row by a number and then det(A) = (-1)^r det(U) where r is the number of times the operation of interchanging 2 rows is used) and gave examples. Showed det(A) is not 0 iff A is invertible and det(AB) = det(A)det(B).
W: Section 3.3: just the part involving determinants as volumes and how linear transformations change volumes (Cramer's rule not covered).
F: Section 5.1:
definitions of eigenvalues, eigenvectors, eigenspaces; examples of how to verify if a value \lambda is an eigenvalue of A and a vector v is an eigenvector of A.

Week 9
M: showed eigenspaces are subspaces, showed eigenspaces corresponding to distinct eigenvalues are linearly indepenedent, and explained how to compute eigenvalues of a matrix A (solve equation det(A-\lambda I) = 0).
W: gave examples of how to compute eigenvalues and bases for eigenspaces of a 2x2 and a 3x3 matrix. Defined algebraic and geometric multiplicity of an eigenvalue. Showed a diagonal matrix has eigenvalues the entries along the diagonal. Showed that a matrix is invertible iff 0 is not an eigenvalue.
F: Thanksgiving. No class.

Week 10
M: Defined similarity and diagonalizability; stated and proved theorem that characterizes when A is diagonalizable; stated steps for diagonalize A.
W: Diagonalization of matrices whose eigenvalues are complex numbers.
F: Show that if A is 2x2 with complex eigenvalues then A = Q C Q^-1 for some Q and C: a rotation matrix. Discuss how to solve initial valued problems of the form: x'(t) = A x(t), x(0)= [a,b] where x(t) is of the form [x_1(t), x_2(t)] and x'(t) =[x'_1(t), x'_2(t)] and a, b are some fixed numbers and A is a fixed 2x2 matrix.