Math 221, Spring 2009

Introduction to Linear Algebra

John Zweck

Course Materials

Schedule and Homework (Updated Daily)
Syllabus [pdf]

Lay's "How to Study Linear Algebra"
Web site for Lay's "Linear Algebra"

Exams from Math 221H Spring 2006

These exams were for an Honors course and so are somewhat more theoretical than the exams will be for this course.

Exam One [pdf]
Solutions to Exam One [pdf]
Exam Two [pdf]
Solutions to Exam Two [pdf]
Final Exam [pdf]

Compulsary Homework Assignment

Write a 2 paragraph statement (no more than 1/2 a page) discussing your study habits. The first paragraph should detail the study habits/skills you have been using for Math and/or related courses and comment on their effectiveness. The second paragraph should state what improvements you will make to your current study habits that will enable you to succeed in Linear Algebra. The more specific and realistic you can be, the better.
To spur your thinking, I suggest you read comments by previous students of mine collected at my Study Habits Survey as well as the syllabus for the course.

The assignment is due at the start of class next Tuesday Feb 10th.

Although it will not count towards your final homework grade, if you do not (eventually) turn it in to me, NONE of your homework will be counted towards your final grade. (Homework assignments of all students will however be graded and returned.) I reserve the right to ask students to revise their statement before I note in my gradebook that you have completed the assignment.

The purpose of the exercise is for you to reflect and strategize. Please avail yourself of this opportunity.

Info Re Quiz

The Quiz on Tu Feb 24 will cover Sections 1.1-1.5 and 1.7. Solutions to the Quiz [pdf]

Info Re Exam One

Exam One on THURSDAY MARCH 5th will cover Sections 1.1-1.5, 1.7-1.9. It will NOT cover 2.1.
Solutions to Exam I [pdf]

Theory for Exam I.
1.1: The 3 elementary row operations; definition of consistency of linear system;
1.2: Row echelon and reduced row echelon forms of matrix; Definition of pivot position; The forward elimination and back substitution algorithms; free and basic variables; Theorems 1,2
1.3: Vectors and their linear combinations and span; geometric interpretations
1.4: Definition of Ax as a linear combination of the columns of A; Theorem 4; Linearity of Matrix-vector multiplication (Theorem 5)
1.5: Homogeneous equation and result on existence of nontrivial solutions; parametric vector form of solution set; Nonhomogeneous equations; Theorem 6; Method to find parametric vector form of solution set of nonhomogeneous equation
1.6: All the green and blue highlighted results
1.8: Definition of linear transformation; Result that matrix transformations are linear
1.9: Theorem 10: All linear transformations on Rn are matrix transformations; Definitions of 1-1, onto linear trasformations; Theorems 11,12

Info Re Exam Two

Exam Two on THURSDAY APRIL 9th will cover Sections 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2.

You should know the following theory topics. Note: If you need to use a result on hwk or in exams you MUST state the result rather than referring to a page number or theorem number in the book!

2.1: Definition of matrix multiplication on top of page 110; fact the each column of AB is a LC of columns of A using weights from corresponding col of B (page 110, blue box); Row-col rule for computing AB (page 111); properties of matrix multiplication (thm 2 page 113);

2.2: Defn of inverse (p 119), formula for det and inverse of 2x2 matrix (p 119); Theorem on solution of Ax=b when A is invertible (Thm 5 p 120); Theorem 6 parts a,b p121; algorithm for finding inverse (pages 124-125);

2.3: Invertible matrix theorem (p129);

3.1: def of determinant (p 187); general co-factor expansion formulae for dets (Thm 1 p 188);

3.2: Theorem re dets and row ops (p 192); A invertible if and only if det(A) is not 0 (Thm 4 page 194); det(AB) = det(A) det(B) (Thm 6 p 196);

3.3: Area of parallelogram and volume of parallelipiped via dets (p 205); Theorem 10 p 207, area case only.

4.1: def of vector space and subspace; theorem 1 page 221.

4.2: def of null space and column space of a matrix and facts they are both subspaces; method to calculate spanning sets for both.

Solutions to Exam II [pdf]

Info Re Final Exam

The final exam will be held Thursday May 14th 1-3pm in MP 103.
The final exam will cover all the sections covered on Exams I and II together with the following sections:
4.3, 4.4, 4.5, 4.6, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 6.1.
About 50% of the final will be on material covered since Exam II, ie on the sections listed on the previous line.

In addition to the theory topics for Exams I and II you should know the following. Note: If you need to use a result on hwk or in exams you MUST state the result rather than referring to a page number or theorem number in the book!

4.3: Definition of Linear Independence; Definition of Basis for vector space (and subspace); Basis for Column space of matrix (Theorem 6); Result that any spanning set has a subset that is a basis (Theorem 5)
4.4: Definition of coordinate vector relative to a basis and theorem that justifies it's existence (Theorem 7); Result that the coordinate mapping is 1-1, onto linear ytransformation (Theorem 8);
4.5: Theorems 9,10,11,12; Definition of finite dimensionality and dimension of a vector space; Methods to obtain bases for null and column spaces of a matrix and the dimensions of these subspaces
4.6: Definitions of row space and rank of a matrix; Rank and Nullity Theorem; How rank enters into Invertible Matrix Theorem
5.1: Definitions of eigenvalue and eigenvector; Result that set of eigenvectors corresponding to distinct eigenvalues are linearly independent;
5.2: The charateristic equation
5.3: The Diagonalization Theorem (Theorem 5); Method for Diagonalizing a matrix; Results that state when a matrix is diagonalizable (a) if all eigenvalues are distinct or (b) if algebraic and geometric multiplicity of each eigenvalue are same
5.4: Result that if entries of matrix are all real then eigenvalues and eigenvectors come in conjugate pairs;
5.6: How to use basis of eigenvectors to compute limit as k goes to infinity of x_k, where x_k = A x_{k-1}.
6.1: Definition of inner (dot) product; Length of vector in terms of dot product; definition of orthogonality of two vectors in terms of their dot product.

Clarifications on Hwk Grading Policy

I read through your hwk submitted last week (late March). While about 15% of you are doing a good job writing it up, many of you need to put in more detail, especially for T/F and written explanation problems. In particular from now on the standards on hwk will be the same as those on the exams:

(a) You will get ZERO points for T/F questions if you just state an answer (T or F) but give no reasons.
(b) Also if your reasons are wrong or incomplete points will be deducted as appropriate. This applies to all written explanation problems too.
(c) Do NOT quote theorem or page numbers from the book. Rather briefly state the theorem or formula you are using.

The point here is that if you put in more detail you will actually be able to understand what you have done several weeks later while studying for exams. The act of writing out the details also helps you reinforce the material. Finally, it is best that you get practice writing solutions up to the standard expected on exams, since that will make the exam a whole lot easier! In the long term, you will retain and be able to use more linear algebra material in future courses.

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