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

Definitions

• orthogonal sets of vectors in $\RR^n$
• an orthonormal basis (ONB)
• orthogonal matrices, the group $O(n)$ of $n\times n$ orthogonal matrices
• the orthogonal complement $W^\perp$ of a subspace $W$ of $\RR^n$
• orthogonal projections
• orthogonally diagonalizable
• a quadratic form
• positive and negative [semi-]definite quadratic forms, indefinite quadratic forms
• an abstract vector space
• vector spaces of functions, $\Ff(\RR)$, of continuous functions $C(\RR)$, of $k$-times continuously differentiable functions $C^k(\RR)$, of polynomials $\Pp$ and of polynomials of degree at most $k$, $\Pp_k$.
• vector subspace
• Span in an abstract vector space
• linearly [in]dependent in an abstract vector space
• basis of an abstract vector space
• dimension of an abstract vector space
• coordinates of a vector $\vec{v}$ with respect to a basis $\Bb$ of an abtract vector space $V$, $[\vec{v}]_\Bb\in\RR^k$, if $\dim(V)=k$
• a change-of-basis matrix
• linear transformations of abstract vector spaces

Theorems

• The Spectral Theorem
• The Gramm-Schmidt Process
• A theorem describing when a subset of (the vectors in) an abstract vector space is a subspace
• A theorem about the properties of change-of-basis matrices
• The Orthogonal Decomposition Theorem
• The Basis Theorem in abstract vector spaces

Things to know how to do

• check if a matrix is in $O(n)$
• orthogonally diagonalize a symmetric matrix
• build an ONB out of any old basis
• orthogonally project a vector into a subspace
• write down the symmetric matrix corresponding to a given quadratic form, and then diagonalize the form
• check if something is or is not a vector space or subspace
• compute the dimension of an abstract vector space, even if it is $\infty$
• check if vectors in an abstract vector space are linearly [in]dependent
• check if a function between abstract vector spaces is a linear transformation

Format of the test

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 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.