Test I Review Sheet for
Colorado State University, Pueblo
Math 307 — Introduction to Linear Algebra —
Spring 2012
Definitions
You should be able to give careful, complete, formal definitions of — and
examples and know theorems involvng — the following terms:
- norm of a vector
- dot products of vectors
- angle between two vectors
- orthogonal vectors
- row equivalent matrices
- [reduced] row-echelon form (=[R]REF) of a matrix
- elementary row operation (ERO)
- elementary matrix
- [in]homogeneous system
- rank of a matrix
- triangular/diagonal matrix
- Span of some vectors
- linear [in]dependent vectors
- [non]trivial solution/linear combination
- subspace of $\RR^n$
- basis of a subspace
- dimension of a subspace
- coordinates of a vector with respect to a basis
- linear transformation
- the (standard) matrix of a linear transformation
- inverse [invertible] matrix
- nullspace, nullity of a matrix
- transpose of a matrix
- [skew-]symmetric matrix
- matrix multiplication
- null-/row-/column-space of a matrix
- determinant of a matrix
- eigenvalue/vector/space of a matrix
- algebraic/geometric multiplicity of an eigenvalue
- characteristic polynomial/equation
- similar matrices
- diagonalizable matrix
Theorems
Please know careful, complete statements of the following:
- The Basis Theorem
- The Rank-Nullity Theorem
- a theorem relating matrices and linear transformations (Theorems 3.30
and 3.31 from the book)
- The Laplace Expansion Theorem
- a theorem relating eigenvalues to the zeros of the characteristic
polynomial of a matrix (see the blue boxes on p.289 in the book)
- The Fundamental Theorem of Invertible Matrices (best to use version 3,
on p.293 of the book — yes, as much as you can get clearly in your
head)
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
- compute the eigenvalues (and their algebraic and geometric
multiplicities), eigenvectors, and eigenspaces of a given matrix
- manipulate equations with matrices, vectors, scalars, transposes,
eigenvalues, linear combinations, inverses, 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.
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 youself
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.