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Syllabus

HOMEWORK ASSIGNMENTS
Week 1 (1/3-1/7): Review the notion of a field in Appendix C. Sec 1.2: 9, 13, 20. Sec 1.3: 15, 19, 20, 23, 24.
Week 2 (1/10-1/14): Sec 1.4: 2(b), 4(d), 12, 15. Sec 1.5: 2(f), 6, 15. Sec 1.6: 2(c), 12, 15.
Week 3 (1/17-1/21): Sec 1.6 (22,23,24,30,31), Sec 2.1 (6,9(c))
Week 4 (1/24-1/28): Sec 2.1 (4,12,15,16,17)
Week 5 (1/31-2/4): Sec 2.2 (5a,8,10), Sec 2.3 (3,4(c),11,12), Sec 2.4 (2(e),3,4,5,6)
Week 6 (2/7-2/11): Sec 2.4 (14,15,16,17), Sec 2.5 (2(d),5,7)
Week 7 (2/14-2/18): Sec 2.5 (11, 13), Sec 3.1 (5, 6, 8), Sec 3.2 (2(d,g),3,4(a))
Week 8 (2/21-2/25): Sec 3.2 (7,18,21), Sec 3.3 (3(d),7(b),9)
Week 9, Spring Break (2/28-3/4):
Week 10 (3/7-3/11): Sec 3.4 (2(e),4(b),5,7,11,12), Sec 4.2 (9,10)
Week 11 (3/14-3/18): Sec 4.2 (25, 27, 28, 29, 30), Sec 4.3 (11, 12, 13, 14, 15)
Week 12 (3/21-3/25): Sec 4.3 (22(c),24), Sec 5.1 (2(d),3,7,8,9,14,15)
Week 13 (3/28-4/1): Sec 5.2 (2f,3d,7)
Week 14 (4/4-4/8): Sec 5.2 (9, 11, 12)
Week 15 (4/11-4/15): Sec 5.4 (15, 21, 23, 24, 26)

Announcements:

Exam 1 is on Wednesday 2-23, chapters 1-2.
Some remarks about Exam 1: The exam is closed book and calculators are not allowed. You may
bring a one-sided sheet with statements (but no proofs) of 6 theorems (or definitions)
of your choice. I will check these sheets during the test, so make sure not to write more on them than allowed!
The best way to prepare for this exam is to solve the homework problems, and select from the abundant list of other problems from the text.
The problems on the exam (there will be 4 or 5) will be very similar, perhaps one will be the same.
You need to study everything covered in class and be able to apply the concepts and theorems to particular applications.

Here is the minimal list of what you should know:
Sec 1.2: Definition of vector spaces (be able to verify on examples)
Sec 1.3: Subspaces: definition and criterion for subspaces
Sec 1.4: Linear combinations, generating (spanning sets): definitions, know how to verify
Sec 1.5: Linear (in)dependence: definition, be able to determine in examples.
Sec 1.6: Basis: definition, representation property (thm 1.8), finite generating sets contain a basis,
dimension of a vector space, procedure to find a basis from a generating set, and extend a linearly
independent set to a basis, the overview at the end of this section as we discussed in class is very important.
Sec 2.1: Linear transformations: definition and knowing how to test for linearity, null space and range
as subspaces (their dimensions nullity and rank), the range is generated by the image of basis vectors,
dimension theorem, 1-to-1 and onto linear transformations (criteria and how to test on examples),
existence of a unique linear transformation that maps a basis in a prescribed set of vectors.
Sec 2.2: Matrix representation of linear transformations (know how to find such matrices,
given ordered bases), the vector space of linear transformations between vector spaces and its properties.
Sec 2.3: Composition of linear transformations (they are linear; main properties)
matrix representation of a composition, relation to matrix multiplication, properties of matrix multiplication
left-multiplication transformation and its properties.
Sec 2.4: Invertible linear transformations (definition, properties, know how to verify for examples),
relation between invertible linear transformation and invertibility of their matrix representation.
Isomorphisms between vector spaces (theorem 2.19 is crucial)
Sec 2.5: Change of coordinate matrices: definition, know how to compute, effect of change of
coordinates on matrix representations of linear transformations.
Class on Tuesday 4/29 is canceled.
Office hours on 4/28 and 4/30 are canceled.
Exam 2 is on Tuesday 4-12, chapters 3-4.
There will be a review session on the material of Exam 2 on Friday 4-1. As usual, this means that you can ask me anything about the material (both theory and exercises). Please prepare accordingly.
The mandatory course/teaching evaluation is scheduled on Monday April 4.
Here is a list of what you should know for Exam 2:
Sec 3.1 : Elementary matrix operations and elementary matrices, invertibility of elementary matrices.
Sec 3.2: Rank of a matrix and of a linear transformation (definitions), properties of the rank (unaffected
by elementary matrix operations), characterization of therank in terms of max number of lin indep columns (i.e. Thm 3.5), Thm 3.6 and Cor 1, rank(A)=rank(A^T) (Cor 2), Cor 3, Thm 3.7,
inverse of a matrix (know how to compute starting from an augmented matrix).
Sec 3.3 : homogeneous systems and properties of their solution sets (Thm 3.8), non-homogeneous systems
and characterization of solution sets (Thm 3.9), and criterion for consistency (Thm 3.11)
Sec 3.4: reduced row echelon form (definition, and know how to bring a matrix in this form in examples,
that is, know Gaussian elimination), solving a system once Gaussian elimination has been performed (Thm 3.15), application in Thm 3.16.
Sec 4.1: definition and properties of determinants of two-by-two matrices, geomtric interpretation (area of paralellogram).
Sec 4.2: definition and properties of determinants of n-by-n matrices (cofactor expansion along ANY
row or column), relation between invertibility, rank and determinant.
Sec 4.3: continuation of properties of determinants, determinant of inverse matrix, det(A)=det(A^T),
Cramer's rule.
Sec 4.4: contains no new material, only a summary of the previous sections.
Exam 2 is closed book and calculators are not allowed. You may bring a one-sided sheet with
statements (but no proofs) of 5 theorems (or definitions).
Last class is on Monday April 18. We will review Chapter 5. Please prepare questions on the material.
The final exam is in our classroom on Friday April 29 between 10:00 and 12:00. This is a comprehensive, closed book exam on everything covered in class on Chap 1,2,3,4 and 5. You can bring a one-sided sheet with statements (but no proofs) of 10 theorems (or definitions). Calculators are not allowed.

Regarding Chap 5, here is what you should know for the final exam (for Chap 1-4, see above):
Sec 5.1: definition of eigenvectors and eigenvalues, definition of diagonalizability, definition of characteristic polynomial. Be able to determine all of the above for given linear operators.
Sec 5.2: Thm 5.5 and its corollary (linear operators with distinct eigenvalues are diagonalizable); Thm 5.6, algebraic and geometric multiplicity of an eigenvalue, eigenspace, Thm 5.7, Thm 5.8, Thm 5.9 (contains an important characterization of diagonalizability!), test for diagonalization, NOT: the material on differential equations (or the problem from control systems discussed in class).
Sec 5.4:invariant subspaces, T-cyclic subspaces generated by a nonzero vector (and their properties), Thm 5.21, Thm 5.22, Cayley-Hamilton's theorem for linear operators and for matrices.