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

Definitions

You should be able to give careful, complete, formal definitions of — and examples and know theorems involving — the following terms (listed in only the vaguest of orders) (below $n\in\NN$):
  1. eigenvalues, eigenvectors, and eigenspaces (of a matrix)
  2. characteristic polynomial (of a matrix)
  3. algebraic and geometric multiplicities (of an eigenvalue)
  4. similarity (of matrices)
  5. equivalence relations
  6. diagonalizable matrices
  7. orthogonal (sets of vectors in $\RR^n$)
  8. an orthonormal basis [ONB] (of a subspace of $\RR^n$)
  9. the Kronecker delta $\delta_{ij}$
  10. orthogonal matrices, the set $O(n)$
  11. the general linear group $GL_n(\RR)$
  12. orthogonal complement $W^\perp$ (of a subspace of $\RR^n$)
  13. orthogonal projection on a subspace
  14. the Gramm-Schmidt Process
  15. orthogonally diagonalizable (for a matrix)
  16. a quadratic form
  17. an abstract vector space
  18. notation: $\Pp$, $\Pp_1$, $\Pp_2$, ..., $C^\infty(\RR)$, $C(\RR)$, $C^1(\RR)$, $C^2(\RR)$, ..., $\Ff(\RR)$, also $C([0,1])$, $C^1([0,1])$, etc.
  19. [vector] subspace (of an abstract vector space)
  20. Span (of a set of vectors in an abstract vector space)
  21. linearly [in]dependent (vectors in an abstract vector space)
  22. basis (of an abstract vector space)
  23. dimension (of an abstract vector space)
  24. [in]finite dimensional (abstract vector space)
  25. coordinates with respect to a basis (in an abstract vector space)
  26. the change-of-basis matrix

Theorems

Please know careful, complete statements (and examples, and counterexamples if a hypothesis is dropped or changed, etc.) of the following:
  1. The Fundamental Theorem of Invertible Matrices, Version 3 (4.17)
  2. The Diagonalization Theorem (4.27)
  3. The Orthogonal Decomposition Theorem (5.11).
  4. The Spectral Theorem (5.20).
  5. The Gramm-Schmidt Process (the book's version is 5.15; I gave a different version in class, which is also acceptable [in fact, I prefer it...]).
  6. The Basis Theorem in an abstract vector space (6.9).
The following are results you will not be asked to state, but which are good to know in order to do exercises which are computations or small proofs:
  1. A theorem on the relationship between invertibility of a matrix and whether or not it has 0 as an eigenvalue (e.g., 4.17).
  2. A theorem on the eigenvalues of triangular matrices (like 4.15).
  3. A theorem on eigenvalues of powers and inverses of matrices (4.18).
  4. A theorem on linear independence of eigenvectors corresponding to distinct eigenvalues (4.20).
  5. A theorem on coordinates with respect to an ONB (5.2).
  6. A theorem giving an alternative characterization of orthogonal matrices (5.4).
  7. A theorem on relationship between orthogonal matrices and norms and dot products (5.6).
  8. A theorem on closure properties of $O(n)$ (5.8).
  9. A theorem on basic properties of the "perp" operation on subspaces (5.9).
  10. A theorem on perp relations among rowspaces and nullspaces of matrices (5.10).
  11. A theorem on basic arithmetic (with $0$ and $\vec{0}$, for example) in an abstract vector space (6.1).
  12. A theorem which describes a span as an intersection of subspaces.
  13. A theorem on existence and uniqueness of coordinates with respect to a basis (6.5).
  14. Theorems on basic properties of bases and dimension (6.10 and 6.11).
  15. A theorem on existence and use of the change-of-basis matrix (6.12 and 6.13).
[In fact, any one of the above, or part of, or a special case of, one of the above, might make a nice statement you would be asked to prove as an exercise. So it would be a good idea not only to know the content of the above theorems, but have some idea of their proofs.]

Things to know how to do

  1. compute eigenstuff of a given matrix
  2. diagonalize a matrix, or even orthogonally diagonalize a symmetric matrix
  3. build an ONB of a subspace of $\RR^n$
  4. prove things about orthogonal, symmetric matrices, or ... matrices
  5. find the projection of a vector onto a subspace, and the orthogonal decomposition of a vector with respect to some subspace
  6. write down the symmetric matrix of a quadratic form, and diagonalize the form
  7. prove that something is or is not an abstract vector space, or subspace of and abstract vector space
  8. determine if a set of vectors in an abstract vector space is linearly dependent or independent
  9. work with the span of a set of vectors in an abstract vector space
  10. find a basis and the dimension of an abstract vector space
  11. prove that an abstract vector space is finite or infinite dimensional
  12. find the coordinates of a vector in an abstract vector space with respect to some basis, and change coordinates with a change-of-basis matrix to the coordinates with respect to a different basis

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 A — E 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.

Consult your I3s, that's what they're for!

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 yourself 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.