Review Sheet for Test I, Fall 2015
Colorado State University, Pueblo
Math 207 — Matrix and Vector Algebra with Applications
Terms you should know -- be able to state clearly what they mean, recognize
them in examples, and create your own examples of them:
- vectors and points in $\RR^2$ and $\RR^3$
- initial point and terminal point of a vector
- magnitude $||\vec{v}||$ of a vector $\vec{v}$.
- vector addition: parallelogram law and component form
- vector subtraction: parallelogram version and component form
- scalar multiplication: geometric version and component form
- unit vector
- dot product: component form and version in terms of angle and
magnitudes
- dot product and magnitude
- orthogonal vectors: geometric meaning and how to tell with dot
products
- vector projection, scalar projection, and orthogonal
projection
- vector equation of a line in the plane or in space, and also its
parametric equations
- vector and parametric equations for a plane in space
- normal form of the equation of a plane in space [this material
was not in the course textbook, but there were supplementary readings
linked from the course homework/schedule page which covered it]
- vector spaces $\RR^2$, $\RR^3$, and $\RR^n$: definitions, vector
addition, scalar multiplication, and dot products
- vector subspaces of $\RR^n$
- matrix notation, addition, subtraction, scalar multiplication
- matrix multiplication: when and how defined, properties
(associative but hardly ever commutative)
- the $n\times n$ identity matrix $I_n$ and what its role is in
matrix multiplication
- translating a system of linear equations into one matrix equation and
vice-versa
- how an $m\times n$ matrix $A$ defines a function $f_A:\RR^n\to\RR^m$
by left multiplication
- a linear mapping/transformation/function $f:\RR^n\to\RR^m$
- Fact: for any $m\times n$ matrix $A$, the function $f_A:\RR^n\to\RR^m$ is
a linear transformation; and in fact every linear transformation
$f:\RR^n\to\RR^m$ equals $f_A$ for some $m\times n$ matrix $A$.
- how to find the $f_A$ for a given linear transformation $f:\RR^n\to\RR^m$
- geometric understandings of linear transformations $f:\RR^2\to\RR^2$:
in particular, know the matrices for rotations, reflections,
and projections
Format of the test
There will be problems similar to the homework, mini- and maxi-quizzes. There
will also be some problems which ask you to synthesize the computations and
perspectives of different parts of the material we've covered. There will also
be an opportunity for you to do show off a particularly nice $I^3$ you might
have created so far this term.
Total points on the test will be more than 100, but your maximum score will be
limited to 100; this means that you can choose which parts of the test to do
and still get a perfect score. If you do earn more than 100 points, the extra
points will be kept as course extra credit.
No book or notes will be permitted. Calculators will be permitted, but will
not be necessary. In particular, there will be problems which require you to
do some computations with vectors and matrices which you will have to do by hand
(e.g., by having symbols in the vectors and matrices rather than merely numbers)
— in short, don't rely upon your calculator being able to do all
computations for you, learn the basic vector and matrix arithmetic yourself as
well.