Math 311-101 — Test 2 Review — Summer
I, 2013
General Information
Test 2 (Wednesday, July 3) will have 6 to 8 questions, some with
multiple parts. It will cover sections 4.2, 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 these sets of
notes:
Change of Basis;
Diagonalization; and
Surfaces . Problems
will be similar to ones done for homework or examples done in class or
in the sets of notes. I will have extra office hours on Tuesday
afternoon, 11:45 am-2:30 pm, and Wednesday morning, 9-9:45 am.
-
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.
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Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
Topics Covered
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. (We will only deal with L:V → V.)
- Subspaces associated with a linear transformation: kernel (or
null space), image of a subspace S, L(S), and the range, L(V). Be able
to use a matrix representation for L to find these.
- Change of Basis
- Be able to find transition matrices SE→F that
change coordinates from E to F. (See my notes
on
change of basis.)
Eigenvalue Problems
- Solving eigenvalue problems
- Eigenvalues and eigenvectors. Given an n×n matrix A, be
able to find A's eigenvalues and their corresponding eigenvectors and
eigenspaces. For an eigenvalue λ, the eigenvectors are all
nonzero vectors in the null space N(A − λ
I). The eigenspace for λ is N(A − λ
I). Be able to find a basis for N(A − λ
I). NOTE: 0 is NEVER an eigenvector. However,
an eigenvalue may be 0.
- Given a linear transformation L, be able to find the matrix A
representing L and to use A to find the eigenvalues and eigenvalues
of L. (See the note on
Diagonalization .)
- 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(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.
- 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. See my notes
on
Diagonalization .
Vector Calculus
- Be able to compute the gradient, divergence, and curl. (Section
8.4)
- Be able to compute line integrals directly or via Green's theorem
(Sections 10.1 and 10.2).
- Parameterized surfaces (See my notes on
Surfaces .)
- Given a parameterization 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 parameterizations 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).
- Questions over vector analysis will make up 35% to 40% of the
test.
Practice tests