Course Exams
Note: More information will be added as the exam dates near
Exam 1: Wednesday Feb. 23, in class
The first exam is not too serious (only worth 10% of your final grade), but the
point is to give you a chance to
examine your mastery of the subject before the Feb. 25 drop date.
However, bear in mind that much of the material after the drop date
becomes considerably more abstract than the material before.
Exam 1 will be on the material covered in lecture through Fri. Feb. 18. Namely,
the following:
matrix operations, systems of linear equations, row reduction, linear
transformations (particularly in the plane), images of linear
transformations.
In particular you should be able to do the following:
- Add and multiply matrices of appropriate sizes
- Know basic properties of matrix addition and multiplication; including
when you can and when you cannot add/multiply 2 matrices
- Go between a system of linear equations and the corresponding augmented matrix
- Determine if a matrix to row echelon form/reduced row echelon form
- Row reduce a matrix to row echelon form/reduced row echelon form
- Solve a given system of linear equations
- Given a geometric description for a linear transformation
(e.g., rotation in the plane by 45 degrees, projection to the xz plane in R^3),
write down a matrix for it
- Given a matrix for a linear transformation, explain what it does geometrically
- Determine the range, domain and image for a linear transformation
- Construct a linear transformation with given range, domain and image
The format will include true/false and problems of the above type.
I recommend you begin studying by first reviewing Homeworks 1-3, and doing
practice problems on the above topics. Monday, Feb. 23 will be review.
Please bring any questions you have to class or office hours
(11:30-12:30, 1:30-2:30) on Monday.
Exam 2: Friday April 15, in class
Exam 2 will be on the material covered in lecture through Mon. April 11,
with a focus on the topics covered since Exam 1. However the material is
cumulative, and you will still need to know the material from the beginning
of the course.
Specifically, the following topics will be covered:
vector spaces, subspaces, span, linear independence, bases, dimension,
coordinates, the image and kernel of linear transformations, rank, nullity
and the Rank-Nullity Theorem.
In particular you should be able to do the following:
- State the definition of all of the concepts listed above
(with the exception of all the axioms defining a vector space)
- Determine if a given space is a vector space
- Determine if a set is a subspace of a given vector space
- Determine if a set of vectors spans a given space
- Determine if a set of vectors is linearly independent
- Find a basis for a space/determine if a set is a basis
- Determine the dimension for a space
- Given a basis S and a vector v, write the coordinate vector [v]_S
- Conversely, given [v]_S, determine the vector v
- *Given a linear transformation, find its image, kernel, rank
and nullity
I have made a sheet of
Practice problems (4/11: Corrected statement of #16
and #24.)
which will similar to the type
of problems to appear on the exam. I recommend you attempt these
before class Wednesday, check your answers with the
Practice problem solutions (4/13: Corrected answer to #15)
then bring any questions you have to class or office
hours. (Please let me know if you spot any errors.)
Final Exam: Friday May 13, 1:30-3:30pm
The final exam is cumulative, so please review the topics listed under Exams 1
and 2. I expect a little more than half of the exam will focus on topics
covered since Exam 2 (though the solution of these problems often requires
understanding of previous material), which are: transition matrices, matrix
with respect to a basis, determinants, inverses, eigenvalues, eigenvectors,
diagonalization and exponentiation.
Specifically, you should be able to do the following:
- State the definitions of eigenvectors and eigenvalues.
- Find and use transition matrices to change bases.
- Write down the matrix of a linear transformation with respect to a
given basis.
- Find the determinant of a matrix.
- Invert a matrix, or determine that it is not invertible.
- Find the eigenvalues and associated eigenvectors for a matrix.
- Diagonalize a matrix, or determine that it is not diagonalizable.
- Exponentiate a matrix.
To help you prepare, I have composed a sheet of
Practice Problems
which I encourage you to treat as a mock exam (prepare first, and try to do
as much as you can on your own, though not necessarily in one sitting, and
probably not under 2 hours).
Check your answers against the
Practice Problem Solutions
You can bring any questions you have to
office hours (Thursday, 2-4pm). Feel free to ask questions by email
or set up an appointment with me.