Final Exam Review Sheet for
Colorado State University, Pueblo
Math 307 — Introduction to Linear Algebra — Fall 2013

Definitions

• 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

• 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

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 Tuesday and Wednesday, both days 10:30am-12:50pm in our usual classroom. Tuesday's test will have the definitions and theorem statements, along with another few exercises; Wednesday's will be entirely exercises (which means mostly proofs), and you will have the definitions from the Tuesday test to look over as you work on Wednesday. Therefore, your memorization (if that is your approach) of definitions and statements should be timed to peak for Tuesday, while Wednesday you should be in top theorem-proving, problem-solving form.

How to study

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.