Study Guide for the 2270-1 final exam Spring 2010

           Introduction to Linear Algebra 2270-2
                     Final Exam Spring 2010


Instructions. The exam is designed for 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. Extra time is scheduled, because of night class conditions.

There is no other day for the exam. If you cannot attend at the
university's scheduled time, or you have 3 exams that day, then please
communicate the problem 581-6879.

Each problem represents several textbook problems numbered (a), (b),
(c), ... . Please solve enough parts to make 100% on each chapter.

Calculators, books, notes and computers are not allowed.

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

Please submit one stapled package of problems, in chapter order.

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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%] Ch3(d): Subspace proof. Function spaces. Sequence spaces.

[40%] Ch3(e): Subspace proof. Hybrid vector spaces.

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

[30%] Ch4(a): Prove or give a counterexample: V is a subspace of W.

[40%] Ch4(b): Find the image, kernel, rank and nullity of T.

[30%] Ch4(c): Find the image and kernel of T.

[40%] Ch4(d): Find a basis for V.

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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

[30%] Ch5(a): Find the orthogonal projection of v onto
V=span(v_1,v_2,v_3).

[40%] Ch5(b): Proof. Pythagorean theorem, CSB-inequality, orthogonal
complement.

[30%] Ch5(c): Find the set of all f in V orthogonal to g.

[30%] Ch5(d): Find the Gram-Schmidt orthonormal vectors.

[40%] Ch5(e): Find the QR factorization of A.

[40%] Ch5(f): Proof. Normal equation for least squares.

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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) by cofactors.

[50%] Ch6(b): Find det(A) using the triangular rule, swap, combo and
multiply.

[50%] Ch6(c): Determinant product theorem and elementary matrices.

[50%] Ch6(d): Inverses using the adjugate formula and also frame
sequence methods. Cramer's rule proof from the adjugate formula.

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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): Solve a discrete dynamical system x(n+1)=A x(n).

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

[50%] Ch7(c): Proof. Subjects are diagonalization, Fourier's model, and the
decomposition AP=PD.

Ch8. (Symmetric Matrices, Singular Value Decomposition)
% Chapter 8: 8.1-6, 8.1-10, 8.1-28, 8.2-12, 8.2-16, 8.2-20, 8.3-6 [example 4]

[50%] Ch8(a): Find an orthonormal eigenbasis for a 3x3 symmetric matrix.

[50%] Ch8(b): Find the equation of a given ellipse in standard corrdinates, 
   displaying the change of basis matrix P and the semiaxes lengths.

[50%] Ch8(c): Find the singular value decomposition of a 2x2 matrix A.
              Illustrate in a figure the meaning of the bases and singular
              values.

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End of S2010 study guide for 2270-2 final exam.