Math 311-101 — Test 2 Review — Summer
I, 2012
General Information
Test 2 (Tuesday, June 26) will have 5 to 7 questions, some with
multiple parts. It will cover sections 4.1-4.3, 5.4, 5.5 (skip bottom
of p. 273 to top of p. 279), 6.1, (skip 6.2), 6.3 in Leon's Linear Algebra (Part
I), plus the notes on
change of basis and
least-squares
problems. Please bring an 8½×11
bluebook. Problems will be similar to ones done for
homework. I will have extra office hours on Monday afternoon, 11:45
am-4 pm, and Tuesday morning, 8:30-9:30 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.
-
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.
- Simple properties. (p. 193)
- 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 find these space for examples similar to ones in the text. The best
way to get these is to use a matrix representaion for L. (See below.)
- 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.)
- Similarity
- 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.)
- Change of basis for a linear transformation. Be able to find the
matrix B of a linear transformation, relative to a basis F, given
that you know its matrix A relative to a basis E. (See my notes
on
change of basis.) Know the definition of similar matrices.
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
on Rn.
- Be able to work with inner products of the form < f,g > =
∫ab f(x)g(x)dx.
- Be able to state Schwarz's inequality and the triangle
inequality.
- 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.
- 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
sqaures approximations for functions. (See my notes
on least-squares problems.)
Eigenvalue Problems
- Solving eigenvalue problems for an n×n matrix A
- Definition. A scalar λ is an eigenvalue of A if
and only if there is a non-zero vector
x for which Ax = λx. The vector x
is an eigenvector associated with λ. Be able to find the eigenvalues and eigenvectors for a matrix, including cases where the eigenvalues are complex numbers.
- Characteristic polynomial.
- pA(λ) =
det(A-λI). Know that pA is a polynomial of degree
n and that
pA(λ) = (-1)n(λn -
trace(A)λ + ... (-1)n det(A)),
where trace(A) = a1,1 + ... + an,n.
- Know that eigenvalues of A are the same as the roots of
pA.
- If A and B are similar, then pB(λ) = pA(λ). Thus, similar matrices have the same eigenvalues.
- Eigenvectors. For an eigenvalue λ, the eigenvectors are all nonzero vectors in the null space N(A − λ I). Finding a basis for N(A − λ I) is what's required. NOTE: 0 is NEVER an eigenvector.
- Diagonalization
- Definition. An n× matrix A is diagonalizable and only if A is similar to a diagonal matrix D; that is, D = S−1AS.
- D = diag(λ1, …, λn), where the λ's are the eigenvalues of A.
- S = [x1 … xn], where xk is the eigenvector corresponding to λk. The matrix S must be invertible.
- Non diagonalizable matrices. There are matrices that can't be diagonalized; for example, the shear matrix A =
- Be able to determine whether or not A is diagonalizable.
- A is diagonalizable if and only if one may extract a basis for Rn (or Cn) from among the eigenvectors of A.
- Know that if the eigenvalues of A are distinct, then A is
diagonalizable. The converse is false, however.
- Be able to diagonalize a matrix A, if that is possible.
- Be able to compute the matrix exponential eA.
Updated: 6/23/2012 (fjn)