Math 603-601 - Midterm Test Review -- Fall 2002
Math 603-601 - Midterm Review
General information The exam will have five to seven questions,
some with multiple parts. The test covers everything we've
discussed. You will be asked to state basic definitions (15 points),
to do simple derivations or proofs (some choice will be given here -
25 points), and to do problems similar to ones given for homework (60
points) or done in class. A list of topics we have covered so far is
given below. Please bring an 8½×11 bluebook.
Vector spaces
- Examples: subspaces, Rn, spaces of functions,
and so on. Be able to determine whether or not a nonempty
subset of a vector space is a subspace.
- Span, linear independence, linear dependence. Be able to
determine whether a set of vectors is linearly independent or linearly
dependent, and whether it spans a given space.
- Basis and dimension. Be able to define these terms. Be able to
determine whether or not a set of vectors is a basis.
- Coordinates and isomorphism. Be able to find coordinate vectors
relative to a given basis. Know what an isomorphism is, and that the
correspondence between a vector space V and Rn
Cn is an isomorpism.
- Dual space. Know the definition of dual space and dual basis.
- Change of bases. Be able to find the change-of-basis matrix, and
be able to change coordinates. Know how the dual basis changes when
the basis in the original space is changed.
Inner products and norms
- Inner product and norm. Be able to define these terms.
- Schwarz's inequality and the triangle inequality Be able to prove
Schwarz's inequality and be able to state the triangle inequality.
- Angle and length. Be able to find the norm of a vector and to
find the angle between two vectors.
- Orthogonality. Be able to define these terms: orthogonal and
orthonormal sets; orthonomal bases; orthogonal matrices.
- Euler angles. Know what the angles precession, nutation, and pure
rotation are. Be able to find the orthogonal matrix corresponding to a
set of these angles.
- Gram-Schmidt. Be able to use the Gram-Schmidt procedure to turn
find an orthonormal set of vectors, given a linearly independent set.
- QR-factorization. Be able to find the QR factorization of a
matrix
- Least squares. Know the difference between continuous and
discrete least squares problems. Be able to do least-squares
approximation using normal equations
Approximation in function spaces
- Series of orthogonal functions. Be able to define what
convergence in the mean is, and know how it is related to Parseval's
equation.
- Fourier series. Be able to find the Fourier series of a given
function.
- Legendre polynomials. Be able to obtain the first few of these
using the Gram-Schmidt process on {1,x,x2,...} in the inner
product
Linear transformations
- Linear transformations. Know the definition, and be able to define
these terms: domain, image, co-domain, null space.
- Matrix representation. Be able to find the matrix of a linear
transformation. Be able to use the matrix representation to change a
linear transformation problem into a matrix problem.
- Similarity and change of basis. Be able to find the new matrix
representing a linear transformation by means of changing basis. Be
able to define similarity; know how this relates to changing bases.
Eigenvalue problems
- Eigenvalues and eigenvectors. Be able to find the eigenvalues and
eigenvectors for a linear transformation or matrix.
- Characteristic polynomial. Be able to find pA(µ)
for a given matrix A. Know the structure of pA. Know that
if B is similar to A, then pB = pA.
- Diagonalization. Define this for a linear transformation. Be able
to diagonalize a matrix or to explain why it cannot be done. Be able
to show that if A is an n×n matrix and if pA has n
distinct roots (real or complex), then A is diagonalizable.
- Selfadjoint matrices. Know that the eigenvalues of a selfadjoint
matrix are real, and that eigenvectors corresponding to distinct
eigenvalues are orthogonal, and that a selfadjoint matrix can always
be diagonalized by an orthogonal matrices.
- Linear systems and normal modes. Be able to solve simple linear
systems of ODEs. Be able to find the normal modes of a spring system.