Math 311101 — Final Exam Review — Summer
I, 2016
General Information
The final exam will be held on Tuesday, July 5 in BLOC 113 (next door
to our usual classroom), from 10:30 am12:30 pm. The exam will have 6
to 8 questions, some with multiple parts. It will cover sections 3.5,
4.14.3, 5.15.3, 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, 11:451:45 and on Friday, 11:4512:45.

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.
Topics Covered
Change of Basis
 Coordinate vectors. Understand how to find coordinates
relative to a basis. See my notes on
Coordinate Vectors
 Transition matrices. Be able to find transition matrices
S_{E→F} that change coordinates from E to F. See my
notes
on
change of basis.
Linear Transformations
 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
operators.
 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
> = y^{T}x is an inner product
on R^{n}.
 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 nonzero 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.
Eigenvalue Problems
 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).
 Diagonalization
 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(λ_{1}, …, λ_{n}),
where the λ's are the eigenvalues of A.
 X = [x_{1} … x_{n}],
where x_{k} is the eigenvector corresponding to
λ_{k}. The matrix X must be invertible.
 For a diagonalizable matrix $A$, be able to find the matrix
exponential $e^A$.
 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.
Vector Calculus
 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).
Practice tests
Updated: 6/28/2016 (fjn)