Final Review Sheet for
Colorado State University, Pueblo
Math 307 — Introduction to Linear Algebra —
Spring 2011
Definitions
You should be able to give careful, complete, formal definitions of — and
examples and know theorems involving — all the terms mentioned in the
reviews for the two midterms, along with the following new ones:
- inner product
- L2 inner product
- inner product space
- orthogonal (vectors in an inner product space)
- norm
- L∞-norm
- normed space
- metric (distance function)
- taxicab metric
(In the extra Sunday review, we discussed how the following were the terms
which we used most frequently during the rest of the term, so are most
important to know:
- linear transformation
- linearly [in]dependent
- vector space
- eigenstuff (you know, —value, —vector, and —space)
- [orthogonally] diagonalizable
- basis
- orthonormal
- kernel
- range
- rank
- nullity)
Theorems/Results/Techniques
Please know careful, complete statements of and/or how to apply the following:
- Orthogonal projection in an inner product space
- Gram-Schmidt in an inner product space
- The Pythagorean Theorem
- The Triangle Inequality
- The Cauchy-Scwarz-Buniakovsky Inequality
- Theorems (e.g., Rank-Nullity, Spectral, Basis), results,
techniques, "things to know how to do" (e.g., diagonalize a
matrix, compute eigenstuff, a dimension, a projection, rank, or
nullity), etc., from previous review sheets.
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 approximately six out of
a choice of a dozen terms. The terms will certainly include some of the recent
ones, plus some of the others we have used most often all semester. Note you
should be able to give context, ideas, examples, etc., for each
term you define, as in the forms from the review sheet for the first test.
Also, you will be asked to give a specific, concrete example for each term,
and a theorem which uses the term.
You may also be asked to give full and formal statements of important theorems
we have used again and again all semester, as well as an example (or
counterexample when some hypothesis is dropped) to which the theorem applies.
You will also be asked to do a few computational tasks — again, things
we have done most often this term, plus something using the most recent class
content — and a few straightforward proofs.