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\examtitle{MATH 2270-003}{Final Exam Guide}{Fall 2010}

\studentinfo
\instructions{You may not use a calculator. Your solutions must include enough justification that another person could understand
 and be convinced by your argument.\\
There are extra blank pages at the end of the booklet. If you need more room to work a problem please note the page number where your work continues.\\}
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The Final Exam is Tuesday, December 14, 10:30-12:30, in the usual
classroom. The exam covers Chapters 1-7 of the text, excluding Section
6.3.

The exam will have two parts, each worth 50\% of the total. Part I
will be similar to a midterm exam. Part II will be an essay
question. You may divide your time between the two parts as you see fit.

\part{}

Direct Computations ($\sim 6$ questions) Topics include:
length of a vector, angle between two vectors, orthogonality, vector
and matrix algebra, solving systems of linear equations, Gaussian
Elimination, Gauss-Jordan Elimination, vector subspaces, basis and
dimension, linear independence, four fundamental subspaces, rank,
projection to a subspace, complementary subspaces, least squares
approximation, Gram-Schmidt process, determinants, eigenvalues and
eigenvectors, similar matrices, diagonalizability, Jordan Form,
Singular Value Decomposition, linear transformations, writing vectors
in different coordinates, representing a linear transformation by a
matrix with respect to chosen bases, pseudo-inverse of a matrix.

For examples, see homework and quiz problems, Sample Exam 1: 1-4, 7,
9, Exam 1: 1-3, 5, 7-8, Sample Exam 2: 1-7, Exam 2: 1-6.

The remaining questions will be different from what you have seen
before, but the style/difficulty of the questions will be similar to
the following:

Indirect Computations ($\sim 2$ questions) A conceptual step reduces
these to direct computation. For examples,
see Sample Exam 1: 5,8, Exam 1: 6, Sample Exam 2: 3, 8, Exam 2: 7-10

Conceptual ($\sim 2$ questions) Conceptual questions and/or
proofs. For examples, see Sample Exam 1: 6, 10, Exam 1: 4, 6, 9,
Sample Exam 2: 9, 10, Exam 2: 8, 9


\part{}
The essay question will cover the Fundamental Theorem of Linear
Algebra and the SVD Theorem. Together, these are the heart of the
course; show me that you understand them. You should think about this
question before the exam, but you will have to write down your answer
during the exam period without using notes.
The following will appear verbatim:

State the Fundamental Theorem of Linear Algebra, Parts 1 and 2, and
the Singular Value Decomposition (SVD) Theorem. Explain, in plain English,
what they mean and how they complement each other. Give an explicit
(actual numbers in the entries)
example of a matrix $M$ such that each of the four fundamental
subspaces is nontrivial, and such that the two bases in the SVD are
not the same. Compute the SVD of $M$ and identify orthonormal bases for
the four fundamental subspaces.




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