Colorado State University, Pueblo; Fall 2015
Math 307 — Introduction to Linear Algebra
Final Exam Review Sheet

Prior Material

Definitions

• linear transformation between abstract vector spaces
• the zero and identity transformations
• 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
• the matrix of a linear transformation between abstract vector spaces with respect to bases of those vector spaces
• 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
• the best approximation in a subspace of a vector
• a least squares solution to a linear system

Theorems

• a theorem describing when finite-dimensional vector spaces are isomorphic
• 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
• a theorem relating whether a linear transformation between finite dimensional abstract vector spaces of the same dimension is 1-1 to whether it is onto
• the Pythagorean Theorem in an inner product space
• The Gramm-Schmidt Process in an inner product space
• The Best Approximation Theorem
• The Least Squares Theorem

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
• compute the matrix of a linear transformation between abstract vector spaces with respect to bases of those vector spaces
• check is something is a valid inner product
• orthogonally project a vector into a subspace of an inner product space
• do Gramm-Schmidt in an inner product space
• find the least squares solution to a linear system
• use the least squares solution to a linear system in applications, such as finding the best polynomial approximation to a collection of points

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 Thursday from 8am to 10:20am and Friday from 10:30am to 12:50pm, both days in our usual classroom. Thursday's test will ask for definitions, theorem statements, and examples, along with one computational exercise for relaxation; Wednesday's will be entirely exercises (which means mostly proofs), and you will have the definitions from the Thursday test to look over as you work on Friday. Therefore, your memorization (if that is your approach) of definitions and statements should be timed to peak for Thursday, while Friday you should be in top theorem-proving, problem-solving form.

Each day of the exam will have around 300 points available, but a maximum allowable score of 200 — extra points to count as course extra credit. Your total final exam score will be the average of your two days' scores.

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. If you can do all HW and maxiquiz problems quickly and clearly, you will be in a very good position for the final.

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 yourself 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.