Final Exam study guide
- Bring your student ID to the final.
- This exam will be covering lectures 1-27 with emphasis
on lectures 21-27
(April 11-May 2) as well as the second half of lecture 20.
These lectures correspond to Sections 6.4 through 8.6.
- The most important concepts and techniques are
- Basic matrix algebra: Properties of matrix operations (especially multiplication), inverses (such as Theorem 1.4.5),
elementary row operations, reduced row echelon form, expressing systems
of equations in matrix form and solving them.
- Determinants: Theorems 2.3.3 and 2.3.4, Figure 2.4.2, row operations,
cofactor expansion, adjoint.
- Vector spaces (Rn in particular): axioms, subspaces, span, linear independence, basis, dimension, and coordinates.
- Vector spaces associated with a matrix: null space, row space, column space, nullity, rank, implications for solving systems of linear equations.
- Inner products (especially the dot product): norm (length) and angles, orthogonal and orthonormal bases (Gram-Schmidt),
orthogonal complements and projections (including least squares approximations),Theorem 6.2.6.
- Dimension formulas: Theorems 5.6.3, 8.2.3, and 6.3.4.
- Linear transformations (especially matrix transformations): kernel (and 1-1), range (and onto), isomorphisms,
matrix of a transformation, change of basis.
- Eigenproblems: determining eigenvectors, eigenvalues,
characteristic polynomials, similarity and diagonalization.
- You will be allowed to use MATLAB on the final.
- You can use a hand written cheat sheet (8.5x11) two-sided to
write down the most important MATLAB commands on the one side and
Theory
(important Theorems, Examples, Definitions, ...) on the other side.
- Hints for what to include in the cheat sheet:
- Theorem *
- The PROPER definitions of all the concepts and objects we have
been working with.
- The CORRECT statements of the most important results we
proved, so that you can apply them correctly.
- The emphasis on the exam will be on understanding of the
subject material not on regurgitation.
- There will definitely be some computational questions.
- There will definitely be some proofs.
- Make sure that you understand the homework problems, since
they give a pretty good cross section of what you have to know.
- If you have the time for it, then doing extra problems
from the book is the way to go. The more you work with the concepts,
the "readier" you are going to be. Here are some sample problems, at least one of which will be on the exam (see also the study guides for the first 2 exams):
- page 384-386, #1-4,6,11-17
- page 446-449, #1-17,20
- Some problems on the exam may throw you off at first.
Make sure that you get a good nights sleep, so that you are fresh,
rested and ready to go!