Test II 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 involvng — the following terms:
- orthogonal ...
- set
- basis
- matrix
- complement
- projection
- decomposition
- orthonormal ...
- orthogonally diagonalizable
- quadratic form
- (abstract) vector space [this one you should expect to have to know
the general shape of the definition, but not all 37 individual parts
[the remaining definitions all are in the context of an abstract
vector space, although there may have been similar definitions for
Rn]
- subspace
- spanning set/span
- linear [in]dependent
- basis
- dimension
- coordinates of a vector with respect to a basis
- change of basis matrix
- linear transformation
- composition, inverse, invertible
- 1-1 or injective; kernel; nullity
- onto or surjective; range; rank
- bijective; isomorphism; isomorphic
Theorems
Please know careful, complete statements of the following:
- The Spectral Theorem
- The Basis Theorem
- The Rank-Nullity Theorem
- A theorem relating the injectivity of a linear transformation to its
kernel.
- A theorem telling when two finite-dimensional vector spaces can be
isomorphic.
Examples
Please know what the following notations mean, and something about how to work
with them:
- C[0,1], Ck[0,1]
- P, Pn
- Mm×n
- O(n)
- [v]B
Things to know how to do
- prove that something is or is not a vector space
- prove that something is or is not a subspace of a given vector space
- prove that something is or is not a linear transformation
- prove that a linear transformation is or is not an isomorphism
- prove that something is a basis
- compute the dimension of a subspace
- compute the kernel/range/rank/nullity of a given linear transformation
- diagonalize a symmetric matrix
- compute an orthonormal basis, starting with some particular vectors
- build an orthogonal projection transformation for a given subspace
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). 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.
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 also be asked to do a small number of the things listed in the "Things
to know how to do" section, above, for particular given data. None of these
will be particularly long or complex.
A take-home part to accompany this test will be handed out on Friday, due back
to me on Monday, on which you will have to do perhaps two out of four possible
exercises. These will either again be of the "Things to know how to do" sort,
but slightly longer, or will be similar (in content and topics) to the more
"proofy" of the homework exercises you have done.