Math 423 - Test I Review
General Information
Test I (Tuesday, February 19) will have 5 to 7 questions, some with
multiple parts. Paper will be provided. Problems will be similar to
ones done for homework. You may use calculators to do arithmetic,
although you will not need them. You may not use any
calculator that has the capability of doing either calculus or linear
algebra.
Topics Covered
- Oscillatory systems and Least Squares - § 2.5, 2.6.
- Be able to set up and solve simple spring mass systems.
- Be able to find the best straight line fit to given data using
least squares.
- Solving linear systems and row reduction - § 3.1-3.4
- Be able to row reduce a matrix. Know what leading entries, rows,
and columns are.
- Be able to solve linear systems via row reduction.
- Elementary row operations and elementary matrices - §
3.5, 3.7
- Know the connection between elementary row operations and the
elementary matrices corresponding to them. Be able to show these
matrices are invertible.
- Be able to do an LU factorization, keeping track of elementary
matrices involved.
- Vector spaces and subspaces - § 5.1, 5.2
- Know the definition of a subspace, and the conditions for a subset
to be a subspace.
- Know the basic examples of vector spaces we discussed -
Rp, C p, P n
(polynomials of degree less than n), and C[a,b], the continuous
functions on [a,b].
- Linear independence, linear dependence, spans - § 5.3
- Know definitions of these. Be able to check sets of vectors to
determine whether they are LI or LD, or whether they span a given
space.
- Bases, dimension, coordinates and coordinate maps - §
5.4
- Know the definitions of the concepts involved.
- Be able to find the coordinates of a vector relative to an ordered
basis.
- Be able to check whether a set is a basis, given the dimension of
a space and other information. For example, if the dimension of a
space is 5, and the set has 7 vectors, it cannot be LI.
- Be able to prove that any p-dimensional vector space is isomorphic
to Rp or C p, depending on whether
it is real or complex.
- Be able to change coordinates - i.e., given coordinates of
a vector relative to one basis, be able to find coordinates relative
to another. Be able to construct change-of-basis matrices.
- Bases for spaces associated with a matrix - § 5.5
- Be able to find bases for the row and column spaces of a matrix.
- Know how to find the rank of a matrix. Be able to show that it is
the same as the row rank and the column rank, which are dimensions of
the row and column spaces, respectively.
- Norms and inner products - § 5.6, 5.7
- Know the definition of a norm and of a real or complex inner
product.
- Three important norms on Rp or C
p - || v ||1, || v ||2, ||
v ||infinity . Given a vector, be able to find these
norms.
- Be able to prove the inequalities showing that these are
equivalent norms.
- Be able to define convergence of a sequence using these norms.
- Be able to prove both Schwarz's inequality and the triangle
inequality for real inner product spaces. Be able to find the angle
between vectors.
- Orthogonal and orthonormal sets of vectors. Be able to show that
an orthogonal set of nonzero vectors is LI.
- Orthogonal projections and Gram-Schmidt - § 5.8.
- Know the definition of an orthogonal projection onto a subspace
V0. Be able to show that if v' is the
orthogonal projection of v onto V0, then we
have
- v - v' is orthogonal to V0.
- v' is independent of the particular orthogonal basis
used to compute it.
- The projection is a linear function of v.
- Best approximation properties.
- Gram-Schmidt procedure.