Study guide for the Math 133 Final Exam
Exam format
The exam format will be similar to the April 2008 exam. It will consist of two sections:
- Multiple choice section. 16 problems worth approx 55% of the exam mark.
- Written response section. 4 problems worth approx 45% of the exam mark.
Exam content
For the exam, you are responsible for all material that has been presented in class,
as well as all material appearing on the written assignments and WebWork. Nothing further.
The course schedule lists which sections from the textbook we covered (note that when an
entire section is listed, e.g 1.1, then we covered all subsections). Yes, there are bits of
subsections that we did not cover, you do not need to know these. No, I'm not going
to try to list all these bits (That's what your class notes are for. I assume that you
came to all the classes.)
Study guide
Here is a list to help you study. You should know about everything on this list.
Disclaimer: This is not guaranteed to be a
comprehensive list. Anything from lectures, WebWork, and assignments
may appear on the final.
Chapter 1 (Linear Equations and Matricies)
- How to simplify matrix expressions involving transpose, inverse, etc. (e.g. Assignment #2, Problem #1)
- How to solve a linear system using augmented matrix and elementary
row operations.
- How to determine how many solutions a linear system has (e.g. Assignment #1, Problems 3 and 4)
- What the rank of a matrix is and how to find it.
- How to find the inverse of an nxn matrix.
- How to determine if a matrix is invertible (Invertible Matrix Theorem).
- What an elementary matrix is and how to write an invertible matrix as a product of elementary matrices.
Chapter 2 (Determinants, Eigenvalues, Diagonalization)
- How to compute the determinant of an nxn matrix.
- The determinant of a triangular/diagonal matrix is easy to compute!
- How elementary row/column operations affect the determinant, and how to use this to 1) simplify
difficult determinant computations, and 2) solve problems like WebWork#4 Problem 12.
- How to simplify expressions involving det, like WebWork #4 Problems 9, 13.
- What the adjoint of a matrix is and what is has to do with the invese (Adjoint Formula).
- How to use Cramer's Rule to solve a linear system.
- The definitions of eigenvalue, eigenvector, eigenspace.
- How to find eigenvalues and eigenvectors.
- The characteristic polynomial, and how it relates to the trace and determinant.
- How to diagonalize a matrix.
- How to use diagonalization to compute (large) matrix powers.
- What it means for matrices to be similar.
Chapter 3 (Vector Geometry)
- The dot product, and how to determine if vectors are orthogonal.
- How length relates to dot product.
- How the dot product relates to the angle between two vectors.
- How to compute the projection of one vector onto another, and how to find the orthogonal component
(also, projections onto a subspace, see Ch. 4).
- Equations of lines and planes.
- The cross product, and how to use it in geometry problems.
- Solving geometry problems, for example: intersection of lines/planes, distances from point/line/plane to
point/line/plane, etc. Some of the important tools here are cross products, projections, and solving linear
systems.
- How to find the area (volume) of a parallelogram (parallelepiped), using cross products/determinants.
- The definition of a linear transformation, and how to check if a transformation is linear.
- How to find the standard matrix of a linear transformation.
- The standard matrices for projection, reflection, rotation in R^2.
- What the image of the unit square under a linear transformation is, and how to find its area.
- The matrix of the composition of linear transformations.
- How to tell if a linear transformation is invertible, and how to find the inverse of a linear transformation.
- What the standard basis of R^n is.
Chapter 4 (The Vector Space R^n)
- The definition of a subspace, and how to show that something is a subspace.
- Know what all the subspaces of R^2 and R^3 are.
- Definition of null space, image space, row space, column space of a matrix.
- What a linear combination of vectors is.
- What the span of a set of vectors is, and how to tell is one vector is in the span of others.
- What it means for vectors to be linearly independent/dependent, and how to show that vectors are
independent/dependent. Note that you should be able to do this both concretely (e.g. WebWork #8 Problem #4) as
well as in a more abstract setting (e.g. Assignment #4 Problem 2(c)).
- The definition of dimension.
- The definition of basis.
- Given a set of vectors, how to find the dimension of their span and a basis for their span
(e.g. WebWork #8 Problem 7) .
- How to use various 'shortcuts' in dealing with dimension and basis (e.g. two vectors cannot be a basis of R^3,
4 vectors in R^3 must be dependent, if U is a subspace of V and dimU=dimV then U=V, 3 vectors in R^3 form a basis iff
the matrix that has them as columns/rows is invertible, etc.).
- How to find bases for rowA, colA, nullA and how the dimensions of these subspaces are related (Rank Theorem).
- What the coordinates of a vector relative to a basis are, and how to find them. In particular, how
to find coordinates relative to an orthogonal basis.
- What an orthogonal/orthonormal basis is.
- The orthogonal complement of a subspace, how to find it, and what its dimension is.
- How to compute a projection onto a subspace, and the orthogonal component (perp_U v).
- How to use the Gram-Schimdt Algorithm to find an orthogonal (orthonormal) basis for a subspace.
- How to orthogonally diagonalize a symmetric matrix.
- How to find eigenvalues/spaces using geometry.
- How to diagonalize a quadratic form.