Final Exam Review Sheet for
Colorado State University, Pueblo
Math 307 — Introduction to Linear Algebra —
Spring 2012
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The final exam will be CUMULATIVE, so the mathematical
continent described in the two previous review sheets, for Midterms
I and II, will
also be on the final and must be studied.
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Definitions
You should be able to give careful, complete, formal definitions of —
and examples and know theorems involving — the following terms:
- kernel of a linear transformation
- range of a linear transformation
- one-to-one and onto
- composition of linear transformations
- inverse of a linear transformation
- isomorphism/isomorphic
- inner product
- inner product space
- $L^2$ inner product on vector spaces of functions
- the norm (length) defined by an inner product
- orthogonal vectors in an inner product space
- projections in an inner product space
- the distance between vectors in an inner product space
- a norm
- a normed linear space
- the distance between vectors in an normed linear space
- the sup/max/$L^\infty$ norm
- the sum/$L^1$ norm
- the taxicab metric
Theorems
Please know careful, complete statements/descriptions of the following:
- a theorem describing when finite-dimensional vector spaces are isomorphic
- The Gramm-Schmidt Process in an inner product space
- a theorem relating the kernel of a linear transformation to whether it is
1-1
- a theorem telling properties of the inverse of a linear transformation
- a theorem telling properties of the composition of linear transformations
- a theorem telling properties of the kernel and range of a linear
transformation
- the Pythagorean Theorem in an inner product space
Things to know how to do
- compute the kernel and range of a linear transformation
- check if a linear transformation is 1-1 or onto
- check if a linear transformation is an isomorphism
- check if two vector spaces are isomorphic
- check is something is a valid inner product
- orthogonally project a vector into a subspace of an inner product space
- normalize vectors in a normed linear space
- do Gramm-Schmidt in an inner product space
- check is something is a valid norm
- draw unit circles in various norms
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 (definitely examples), and other related
components, for each term you define, as in the definition forms handed out in
class.
You will 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.
In short, expect to have to do on the final the things we have done in class
over and over, including (but definitely not limited to):
- state a definition clearly and completely
- give a completely explained example
- state a theorem clearly and completely
- verify a definition in a new or unexpected situation
- apply a known theorem to derive a useful consequence in an example or in
the middle of a proof
- prove a theorem (or at least a proposition) about common objects or
properties we have frequently used
- write down the definitions appearing in a theorem statement, in order
then to put together the hypotheses and derive, perhaps after one (or
two...) small calculations, the conclusion ["proof by unpacking"]
- prove a simple statement by induction
- prove a simple statement by contradiction
- prove a simple statement by going after the contrapositive
- ...
The test will take place on Monday and Tuesday, both days 8:00-10:20am in our
usual classroom. Monday's test will have the definitions and theorem
statements, along with another few exercises; Tuesday's will be entirely
exercises (which means mostly proofs), and you will have the definitions from
the Monday test to look over as you work on Tuesday. Therefore, your
memorization (if that is your approach) of definitions and statements should be
timed to peak for Monday, while Tuesday you should be in top theorem-proving,
problem-solving form.
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 final. Make sure you have finished the HW and
internalized the comments I made on graded HW, maxiquizzes, and midterms.
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 final problem.
You can easily use the above to make a practice final for yourself: give
youself a couple of hours and try to write some nice definitions, nice
theorem statements, and prove several HW or maxiquiz problems or lemmata
from the book.