Math 207 — Matrix and Vector Algebra with Applications

Course Schedule & Homework Assignments

Here is a link back to the course syllabus/policy page.

This schedule is will be changing **very frequently**, please check it at
least every class day, and before starting work on any assignment (in case the
content of the assignment has changed).

In the following all reading assignments, sections, and page numbers refer to
the required course textbook, * Computational Matrix Algebra*, by
William D. Emerson, unless otherwise specified.

If you see the symbol below, it means that class was videoed and you can get a link by e-mailing me. Note that if you know ahead of time that you will miss a class, you should tell me and I will be sure to video that day for you.

When there is a reading assignment, please read the named section(s) **before
that day**.

Homework for a particular day is **due that day**, either in class or
handed in at my office **by 3pm**.

*:*- bureaucracy and introductions.
- Read the course syllabus and policy page.
**HW0**(at*Send me e-mail*`jonathan@poritz.net`) telling me:- Your name.
- Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
- Your year/program/major at CSUP.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math classes you've had.
- The reason you are taking this course.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*). - [Optional:] The name of a good book you have read recently.

- Some content we discussed, terms defined:
**vectors**:**direction**and**magnitude**[=**length**]- when are two vectors
**equal** - the
**initial point**[**IP**] and**terminal point**[**TP**] of a vector - write $(a,b)$ for the point with
**coordinates**$a$ and $b$ - write $\vec{v}$ for a variable which represents a vector, and just $P$ for a variable which represents a point.
- the
**components**of a vector are written $<a,b>$ (these are the coordinates of the TP of a vector if it has been moved so the IP is the origin) - a vector uniquely determines, and is uniquely determined (up to equality) by its components
- a formula, and notation, for the
**length**of a vector $||<a,b>||=\sqrt{a^2+b^2}$. **scalar multiplication**of a vector $\vec{v}$ by a**scalar**[just a real number] $t$, defined by $t<a,b>=<ta,tb>$.- a scalar multiple of a vector, $t<a,b>$, has length which is $|t|$ times the length of $<a,b>$ — in symbols, $||t<a,b>||=|t|<a,b>$; it is in the opposite direction from $<a,b>$ if $t$ is negative.

*:***Read:**§1.1- Some content we discussed, terms defined:
- scalar multiplication with vectors in three dimensions
- lengths of vectors in three dimensions
- unit vectors — how to make any non-zero vector $\vec{v}$ into a unit vector: scalar multiply by $\frac{1}{||\vec{v}||}$.
- vector addition (and subtraction) in two and three dimensions, in terms of components
- vector addition geometrically: the main diagonal of the parallelogram formed by the two vectors you're adding
- vector "subtraction": add the negative of the vector you are
trying to subtract; geometrically, that is the
*other diagonal*of the same parallelogram as for adding the vectors. - defined the dot product of two vectors
- noticed that sometimes $\vec{v}\cdot\vec{w}=0$ even though neither $\vec{v}$ nor $\vec{w}$ is the zero vector -- this is very different from the normal product of two real numbers, where the Zero Product Property applies (The ZPP says that for $a,b$ real numbers, if $ab=0$ then at least one of $a$ or $b$ must $0$).
- for any vector $\vec{v}$, $||\vec{v}||=\sqrt{\vec{v}\cdot\vec{v}}$.

**Miniquiz 1 today**

*:***Read:**§1.2- Some content we discussed, terms defined:
- more algebraic properties of dot products, such as commutativity and distributativity over vector addition
**Theorem**If the angle between vectors $\vec{v}$ and $\vec{w}$ is $\theta$, then $\vec{u}\cdot\vec{v}=||\vec{u}||\,||\vec{v}||\cos{\theta}$.- the fancy way of saying two vectors are perpendicular is to say
they are
**orthogonal** - vectors are orthogonal if and only if their dot product is 0
- The
**projection of $\vec{v}$ onto $\vec{u}$**, which we write $\operatorname{proj}_{\vec{u}}(\vec{v})$, is the vector one gets by dropping a perpendicular from $\vec{v}$ onto the line along $\vec{u}$ - If $\vec{u}$ is a unit vector, then the length of $\operatorname{proj}_{\vec{u}}(\vec{v})$ is $\vec{u}\cdot\vec{v}$. In other words, $||\operatorname{proj}_{\vec{u}}(\vec{v})||=\vec{u}\cdot\vec{v}$.

**Maxiquiz 1 today**- Hand in
**I**and^{3}1**HW1**: 1.1.{6, 8, 11, 13, 17} **Today [Friday] is the last day to add classes.**

*:***[Re]Read:**§1.2- Some content we discussed, terms defined:
- The terms
**scalar projection**,**vector projection**, and**orthogonal projection**

- The terms
**Miniquiz 2 today**

*:***Read:**§1.3- Some content we discussed, terms defined:
- line along a vector
- vector form of the equation of a line
- parametric form of the equation of a line
- vector form of the equation of a plane
- parametric form of the equation of a plane
- vector forms of parametric curves

- Hand in
**I**and^{3}2**HW2**: 1.2.{1, 7, 23, 28} **Miniquiz 3 today**

*:***Read:**This page by your instructor (and/or this page and/or this one and/or this one by other writers ... or just take good notes in class)- Some content we discussed, terms defined:
- Another way to think about $\operatorname{proj}_{\vec{u}}(\vec{v})$: the vector on the line along $\vec{u}$ which is closest to $\vec{v}$.
- lines through the origin normal (perpendicular) to a given vector
- equations of arbitrary lines in normal form
- planes through the origin normal to a given vector
- equations of arbitrary planes in normal form

**Maxiquiz 2 today**

*:***Read:**§1.4**THE BOOK IS IN, AT THE BOOKSTORE! PLEASE GO GET IT!**- Some content we discussed, terms defined:
- dropping the notational difference between $\left<x,y\right>$ and $(x,y)$, or the same thing in three dimensions
- definitions of $\RR^2$, $\RR^3$, and in fact $\RR^n$ for any positive whole number $n$
- vector operations on $\RR^n$
- vector space properties of $\RR^n$; or, the definition of a
**vector space**. - definition of a
**vector subspace**, examples

**Today [Monday] is the last day to drop classes without a grade being recorded.**- Hand in
**I**and^{3}3**HW3**: 1.3.{4, 7, 12} and**Extra problem HW3X1:**Find both the normal and and then the vector or parametric equations for the plane through the point $(-1,-1,-1)$ which is parallel to the plane with normal equation $x+2y+3z=4$.*[Hints: can you read off the normal vector from the given normal equation? if two planes are parallel, how are their normal vectors related? do the normal equation first; once you have that, can you get three points on the plane and use them to figure out the vector or parametric equations of the plane?]* **Miniquiz 4 today**

*:***Read:**§2.1- Some content we discussed, terms defined:
- matrices: definitions, notation
- the diagonal of a matrix, $\operatorname{diag}(A)$
- matrix operations: addition, subtraction, multiplying by a constant
- matrix multiplication: definition, basic properties
(associativity but
**not commutativity!**) - matrix operations and systems of linear equations

**Miniquiz 5 today**

*:***Read:**§2.2- Some content we discussed, terms defined:
- matrices multiplying vectors on both sides
- another way to think of a matrix times a vector, as combining the columns of the matrix
- building a function $f_A:\RR^m\to\RR^n$ out of multiplication on the left by an $m\times n$ matrix A
- properties of these functions $f_A$
- defining
**linear mapping/function/transformation**

**Maxiquiz 3 today**- Hand in
**I**and^{3}4**HW4**: 1.4.5 and 2.1.{1, 3, 4}

*:***Read:**§2.3- Some content we discussed, terms defined:
- finding the matrix $A$ of a linear transformation
$f:\RR^m\to\RR^n$,
*i.e.,*given $f$, find $A$ such that $f=f_A$ - the books "revolutions around the axes," when in the plane, are the same as reflections across those axes
- rotation of the plane by a given angle
- projections as matrix multiplication

- finding the matrix $A$ of a linear transformation
$f:\RR^m\to\RR^n$,
**Miniquiz 6 today**

*:*- Review for Test I. See this review sheet.
- Hand in
**I**and^{3}5**HW5**: 2.2.{1, 3}, 2.3.{2, 6} **Miniquiz 7 today**

*:***Test I in class today.**

*:*- Test I post-mortem.
- Some content we discussed, terms defined:
- remembering representing systems of linear equations as matrix equations
- defining the
**coefficient**and**augmented matrices**of a linear system

*:***Read:**§2.4- Some content we discussed, terms defined:
**Gaussian elimination**with back substitution**Gauss-Jordan elimination****elementary [row] operations**

- Hand in Test I revisions, if you like.

*:***Read:**§2.5- Some content we discussed, terms defined:
- Linear systems with no solutions or an infinite number of solutions
- geometric interpretation of systems of linear equations in two or three variables

**Maxiquiz 4 today**

*:***Read:**§2.6- Some content we discussed, terms defined:
- two matrices being
**row-equivalent** - the process of
**row-reducing**a matrix - a matrix in
**row-echelon form [REF]** - a matrix in
**reduced row-echelon form [RREF]** - outline of
*Gaussian elimination*with back substitution - outline of
*Gauss-Jordan*to solve a system of linear equations - the possible shapes of an RREF version of an augmented matrix
which correspond to a system of linear equations which has has
- a unique solution
- no solution
- infinitely many solutions (free variable(s)!)

- two matrices being
- Hand in
**I**and^{3}6**HW6**: 2.4.2, 2.5.{2, 3} **Miniquiz 8 today**

*:***Read:**§2.7- Some content we discussed, terms defined:
- the
**inverse**$A^{-1}$ of an $n\times n$ matrix $A$ - discussing existence and uniqueness of inverses
- finding the inverse of a matrix $A$ by row-reducing $\left(A\ \,I_n\right)$ to RREF form
- using $A^{-1}$ to solve a system of linear equations with coefficient matrix $A$ [sometimes with many different right-hand sides for the system]
- introduction to computational tools to help with the
row-reducation, such as
`Matlab`,`Octave`, Wolfram Alpha, or a random applet someone put on the 'net

- the
**Miniquiz 9 today**

*:***Read:**§3.1- Some content we discussed, terms defined:
**linear combinations**of $k$ vectors in $\RR^n$- the
**Span**of $k$ vectors in $\RR^n$; a**spanning set** - what it means for $k$ vectors in $\RR^n$ to be
**linearly [in]dependent**

- Hand in
**I**and^{3}7**HW7**: 2.6.{1, 4}, 2.7.{2, 3, 8 [where the book says "*Mathematica*", read "*Matlab*" or "*Octave*" or any computational tool you like} **Maxiquiz 5 today**

*:***[Re]Read:**§3.1 and §3.2- Some content we discussed, terms defined:
- a theorem relating the linear independence of some vectors $\vec{v}_1,\dots,\vec{v}_k$ to solutions of a matrix equation with matrix whose columns are the vectors $\vec{v}_1,\dots,\vec{v}_k$.
- a
**basis**of a vector space - bases exist, but are not unique
- the
**null space**$\operatorname{null}(A)$ of a matrix $A$ - the
**range**$\operatorname{range}(A)$ of a matrix $A$ - if $A$ is $m\times n$ then $\operatorname{null}(A)$ is a vector subspace of $\RR^n$ and $\operatorname{range}(A)$ is a vector subspace of $\RR^m$

**Miniquiz 10 today**

*:***Read:**§3.3- Some content we discussed, terms defined:
- $2\times2$ and $3\times3$
**determinants**and properties

- $2\times2$ and $3\times3$
- Hand in
**I**and^{3}8**HW8**: 3.1.{2, 3}, 3.2.{2, 5} .. ok, you may hand this in on Friday, but it would be a very good idea to get started before then... **Miniquiz 11 today**

*:***Read:**§3.4- Some content we discussed, terms defined:
- cross products of vectors in $\RR^3$

- Hand in
**I**and^{3}8**HW8**: 3.1.{2, 3}, 3.2.{2, 5} **Maxiquiz 6 today**

*:*- Review for Test II. See this review sheet.
- Hand in
**I**and^{3}9**HW9**: 3.3.{2, 3, 5}, 3.4.{1, 4, 11} **Miniquiz 12 today**

*:***Test II in class today.**Bring your calculator, if you like — you may use it during the test

*:*- Test II post-mortem.
- no mini- or maxiquiz today, alas...

*:***Read:**§4.1- Some content we discussed, terms defined:
- eigenvalues and eigenvectors
- characteristic polynomial

- Hand in Test II revisions, if you like.
- no mini- or maxiquiz today, alas...

*:***Read:**§4.2- Some content we discussed, terms defined:
- eigenspaces

- Hand in
**I**and^{3}10**HW10**: 4.1.{1, 4, 6, 9, 12} **Miniquiz 13 today**

*:***Read:**this page by your instructor- Some content we discussed, terms defined:
- Markov models:
- the idea
- the
**Markov hypothesis** - a
**transition matrix** - a
**stable distribution**(also called a "**steady state vector**")

- Markov models:
**Maxiquiz 7**was handed out today: it is due on Monday

*:*- Review for Final Exam. See this review sheet.
- Hand in
**I**and^{3}11**HW11**: 4.2.{3, 4, 5, 7} - hand in
**Maxiquiz 7** **Miniquiz 14 today**

*:*- Drop by my office if you want your
*Maxiquiz 7*and*HW11*back to look over before the final.

- Drop by my office if you want your
*:***Final Exam, Part I**in class today.

*:***Final Exam, Part II**in class today.