Colorado State University, Pueblo; Fall 2015
Math 307 — Introduction to Linear Algebra
Test I Review Sheet
Definitions
You should be able to give careful, complete, formal definitions of — and
examples and know theorems involving — the following terms (listed in only
the vaguest of orders):
- norm of a vector
- dot products of vectors
- angle between two vectors
- orthogonal vectors
- unit vectors
- projection of one vector onto another
- linear combination of vectors, trivial linear combination
- row equivalent matrices
- [reduced] row-echelon form (=[R]REF) of a matrix
- elementary row operation (ERO)
- elementary matrix
- system of linear equations
- solution [trivial solution] of a linear system
- free variable
- [in]consistent system
- [in]homogeneous system
- rank of a matrix
- triangular/diagonal matrix
- Span of some vectors
- linearly [in]dependent vectors
- subspace of $\RR^n$
- basis of a subspace
- the coordinates of a vector with respect to a basis
- dimension of a subspace
- linear transformation
- the matrix $[T]$ of a linear transformation $T$
- inverse [invertible] matrix
- [non-]singular matrix
- nullspace, nullity of a matrix
- transpose of a matrix
- [skew-]symmetric matrix
- matrix multiplication
- null-/row-/column-space of a matrix
- nullity of a matrix
- converse of a logical statement
- contrapositive of a logical statement
- the standard basis of $\RR^n$
- determinant of a matrix
Theorems
Please know careful, complete statements of the following:
- The Basis Theorem (Theorem 3.23)
- The Rank-Nullity Theorem
- a theorem relating matrices and linear transformations (Theorems 3.30
and 3.31 on pp.220-222 in the book)
- The Laplace Expansion Theorem (Theorem 4.1 on p.277 in the book; we are
essentially using this theorem to prove that the determinant as we defined
it in class is well-defined)
- The Fundamental Theorem of Invertible Matrices (best to use version 2,
on p.212 of the book — yes, as much as you can get clearly in your
head)
- theorems on properties of the determinant (see, e.g., Theorems
4.4, 4.6, and 4.8 on pp.282-284 in the book)
Things to know how to do
- prove something by induction
- prove that some set of vectors is linearly [in]dependent
- prove that something is or is not a subspace of $\RR^n$
- prove that something is or is not a linear transformation
- prove that something is a basis
- compute the dimension of a subspace
- compute the rank and nullity of a given matrix
- computer the determinant of a matrix
- manipulate expressions with matrices, vectors, scalars, transposes,
linear combinations, inverses, determinants, etc., to do some
computation, either for itself or as part of a proof.
Format of the test
Please be able to state — in full formality and five-part harmony —
all the definitions. You will be asked to do so for something like eight out of
a choice of ten terms (or maybe ten out of twelve). You must also be able to
give context, ideas, examples, etc., for each term you define, as in the
definition forms handed out in class
You may also be asked to give full and formal statements of theorems mentioned
above, as well as an example (or counterexample when some hypothesis is
dropped) to which the theorem applies. You will certainly have to use those
theorems in proving other statements....
You will also be asked to do several of the things listed in the "Things to
know how to do" section, above. Mostly, this will be in the form of proofs
(smallish, like the maxiquizzes), with perhaps a straight, semi-numerical
calculation thrown in for relaxation.
How to study
You must know the definitions, in the full form as in the forms handed out in
class (or available at
this web page). This is a good place to start.
You should likewise know the theorem statements. See the book for full, formal
statements.
Use flashcards, tattoos, limericks, chanting, or whatever works for you to get
the above into your head.
Consult your I3s, that's what they're for!
The HW exercises and maxiquizzes are similar in content, style, and difficulty
to what will be on the test. Make sure you have finished the HW and
internalized the comments I made on graded HW and maxiquizzes.
Read the proofs in the book. Make sure you have those proofs quite clearly in
your head, both because they will be good models for your own future
proof-writing and because a book proof of a lemma or small but important part of
a big proof would make a good midterm problem.
You can easily use the above to make a practice test for yourself: give yourself
an hour and try to write eight nice definitions, a couple of nice theorem
statements, and prove three or four HW or maxiquiz problems or lemmata from the
book.