Previous, old information, is attached below.
The information printed directly below reflects the actual exam after
construction and testing.

Information on the 2270-1 final exam Spring 2004

           Introduction to Linear Algebra 2270-1
               Final Exam 8:00am 5 May 2004


Instructions. The time allowed is 120 minutes. The examination
consists of six problems, one for each of chapters 3, 4, 5, 6, 7,
8, each problem with multiple parts. A chapter represents 20
minutes on the final exam.

Each problem represents several textbook problems numbered (a),
(b), (c), ... . Please solve enough parts to make 100% on each
chapter. Choose the problems to be graded by check-mark X; the
credits should add to 100. In problem Ch3, work three (Ch3(d)
counts as 30% or 40%). For the others, Ch4 to Ch8, work two.

Calculators, books, notes and computers are not allowed.

Answer checks are not expected or required. First drafts are
expected, not complete presentations.

Please submit exactly six separately stapled packages of problems.

===============================================================
Ch3. (Subspaces of R^n and Their Dimensions)
% Chapter 3: 3.1-31, 3.1-49, 3.2-17, 3.2-33, 3.2-50, 3.3-7, 3.3-32,
% 3.4-13, 3.4-51

[30%] Ch3(a): Find a basis in R^3 for the image of A.

[40%] Ch3(b): Find the matrix of T relative to the basis v_1,
v_2, v_3.

[30%] Ch3(c): Find a basis in R^4 for the kernel of A.

[40% or 30%] Ch3(d): Subspace proof.

===============================================================
Ch4. (Linear Spaces)
%  Chapter 4: 4.1-10, 4.1-29, 4.2-29, 4.3-37

[50%] Ch4(a): Prove or give a counterexample: V is a subspace of W.

[50%] Ch4(b): Find the image, kernel, rank and nullity of T.

[50%] Ch4(c): Find the image and kernel of T.

[50%] Ch4(d): Find a basis for V.

===============================================================
Ch5. (Orthogonality and Least Squares)
% Chapter 5: 5.1-12, 5.1-27, 5.2-13, 5.3-7, 5.3-25, 5.4-5, 5.4-31,
% 5.5-23

[50%] Ch5(a): Find the orthogonal projection of v onto
V=span(v_1,v_2).

[50%] Ch5(b): Prove or give a counterexample.

[50%] Ch5(c): Find the set of all f in V orthogonal to g.

[50%] Ch5(d): Find the Gram-Schmidt orthonormal vectors.

===============================================================
Ch6. (Determinants)
% Chapter 6: 6.1-16, 6.2-13, 6.2-19, 6.3-14, 6.3-25

[50%] Ch6(a): Evaluate det(A).

[50%] Ch6(b): Find det(A).

[50%] Ch6(c): Find the area of the parallelogram formed by v_1,
v_2.

===============================================================
Ch7. (Eigenvalues and Eigenvectors)
% Chapter 7: 7.1-11, 7.1-33, 7.2-9, 7.3-15, 7.3-17, 7.4-15, 7.4-19,
% 7.5-25

[50%] Ch7(a): Discrete dynamical system x(n+1)=A x(n).

[50%] Ch7(b): Find all eigenpairs of A.

[50%] Ch7(c): Proof.

===============================================================
Ch8. (Symmetric Matrices and Quadratic Forms)
% Chapter 8: 8.1-5, 8.1-25, 8.2-3, 8.2-19, 8.3-11, 8.3-17

[50%] Ch8(a): Find an orthonormal matrix S such that
S^{-1}AS is diagonal.

[50%] Ch8(b): Find the ellipse semi-axis lengths a,
b and the unit semi-axis directions v_1, v_2.

[50%] Ch8(c): Find a singular value decomposition
A=U Sigma V^T.

[50%] Ch8(d): Singular value
decomposition proof.
===============================================================
End of new information Sun May  2 12:39:31 MDT 2004


Old information:

The 2270-1 final exam consists of twelve problems, from which you
are expected to complete six for full credit. However, you may
choose only one problem from each of chapters 3 to 8.

The following problems will be used as models for the problems
that will appear on the 2270-1 final exam. Each problem will have
one to five parts, to facilitate division of credit for that
problem.

Topics outside the subject matter of these problems will not be
tested. For instance, you may be asked to prove the triangle
inequality in an inner product space, because that is the subject
of one the problems below. You will not be asked to prove the
Cauchy-Schwartz inequality, because no problem directly references
it.

Any theorem directly used in one the problems below is fair game,
which means you may be asked to provide details of proof for that
theorem. Theorems unreferenced in the problems below will never
appear in a final exam question.

 Chapter 3: 3.1-31, 3.1-49, 3.2-17, 3.2-33, 3.2-50, 3.3-7, 3.3-32,
 3.4-13, 3.4-51

 Chapter 4: 4.1-10, 4.1-29, 4.2-29, 4.3-37

 Chapter 5: 5.1-12, 5.1-27, 5.2-13, 5.3-7, 5.3-25, 5.4-5, 5.4-31, 5.5-23

 Chapter 6: 6.1-16, 6.2-13, 6.2-19, 6.3-14, 6.3-25

 Chapter 7: 7.1-11, 7.1-33, 7.2-9, 7.3-15, 7.3-17, 7.4-15, 7.4-19, 7.5-25

 Chapter 8: 8.1-5, 8.1-25, 8.2-3, 8.2-19, 8.3-11, 8.3-17