Math 311-101 — Final Exam Review — Summer
The final exam will be held on Tuesday, July 5 in BLOC 113 (next door
to our usual classroom), from 10:30 am-12:30 pm. The exam will have 6
to 8 questions, some with multiple parts. It will cover sections 3.5,
4.1-4.3, 5.1-5.3, 5.5, 6.1, 6.3, 8.4 (gradient, divergence, and curl),
10.1, 10.2, 11.2, and 11.3 in the text. In addition, it will include
material from my notes on
Coordinate Vectors and on
Change of Basis. Problems will be similar to ones done for
homework or examples done in class or in the sets of notes. I will
have office hours on Thursday, 11:45-1:45 and on Friday, 11:45-12:45.
Calculators. You may use scientific calculators to do numerical
calculations logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
Change of Basis
- Coordinate vectors. Understand how to find coordinates
relative to a basis. See my notes on
- Transition matrices. Be able to find transition matrices
SE→F that change coordinates from E to F. See my
change of basis.
- Definition and Examples
- Know the definition of a linear transformation. Be able to
determine whether a transformation is linear.
- Matrix Representations
- Know how to find matrix representations for linear
transformations. Be able to work problems similar to ones done in
class and to ones in the homework. Specific examples are matrix
representations for shear transformations, rotations,
dilation/contraction operations, using shear transformations to do
translations via matrix multiplication, differential and integral
- If a matrix $A$ represents a linear transformation $L$ and a
change of basis is made, be able to find the matrix $B$ that
represents $L$ in the new basis.
- Subspaces associated with a linear transformation: kernel (or
null space), and the range, R(L) (also, L(V)). Be able to use a matrix
representation for L to find these.
Inner Product Spaces
- Inner product
- Inner product and norm. Know the definition of an inner product
and its associated norm (length). Be able to show that < x,y
> = yTx is an inner product
- Be able to state and prove Schwarz's inequality. (See
class notes for June 21, 2016.)
- Angle and length. Be able to find the norm of a vector and to
find the angle between two vectors.
- Orthogonal and orthonormal sets
- Be able to define these terms: orthogonal and
orthonormal sets; orthonormal bases.
- Be able to verify that a set is orthogonal or orthonormal.
- Orthogonal subspaces. Fundamental theorem of linear algebra,
Fredholm alternative (notes 6/22/16), direct sum, ⊕.
- Be able to show that sets of non-zero orthogonal vectors are
linearly independent and that that vectors can be represented as in
Theorem 5.5.2 (i.e., be able to PROVE this theorem).
- Least squares. Know how to use an orthonormal set to find least
squares approximations for functions.
- Solving eigenvalue problems
- Given an n×n matrix A, be able to find A's eigenvalues and
their corresponding eigenvectors and eigenspaces.
- Applications. Simple electric circuits and normal modes
in a spring system. (See the notes for 6/28/16).
- Be able to diagonalize an n×n matrix A, if that is
possible. The matrix A is diagonalizable and only if it is similar to a
diagonal matrix D; that is, A = XDX−1.
- D = diag(λ1, …, λn),
where the λ's are the eigenvalues of A.
- X = [x1 … xn],
where xk is the eigenvector corresponding to
λk. The matrix X must be invertible.
- For a diagonalizable matrix $A$, be able to find the matrix
- Non diagonalizable matrices. There are matrices that can't be
diagonalized. These are called defective. Be able to
determine whether A is diagonalizable or defective.
- Be able to compute the gradient, divergence, and curl.
- Be able to compute line integrals directly or via Green's theorem.
- Parametrized surfaces
- Given a parametrization for a surface, be able to find these
quantities: standard normal N, unit normal n, vector
area element dS, and the scalar area element dS.
- Know parametrizations for spheres, cylinders, and planes. For
each of these, know N, n, dS, dS.
- Surface integrals
- Be able to compute both scalar and vector surface integrals,
either directly (Colley, section 11.2), or via Stokes's
Theorem or Gauss's Theorem (Colley, section 11.3).
Updated: 6/28/2016 (fjn)