call # 13420-2
MATH 601, M-W-F 9:30 a.m., SO 0056
A DAY-BY-DAY LIST OF TOPICS
Sept. 24: VECTOR SPACES (real & complex), spaces of arrow vectors, the numerical
spaces IKn, function spaces, subspaces, nite sets of vectors as opposed to unordered
systems and
MATH 602, WINTER 2009
4. COMPLETENESS
Cauchy sequences in metric spaces were dened in 1. Let xj , j = 1, 2, . . . , be a
sequence of points in a metric space (M, d).
(i) If xj converges, then it has the Cauchy property.
(ii) If xj is a Cauchy sequence, th
MATH 602, WINTER 2009
The Bolzano-Weierstrass theorem
The Bolzano-Weierstrass theorem states that every bounded sequence of real
numbers has a convergent subsequence.
Here is a proof. The fact that a real number y has a decimal expansion n.d1 d2 d3 . . .
MATH 601, AUTUMN 2007
Elementary modications
1. Denitions.
Let X be a nite system of vectors in a vector space V over a eld K. Another such
system X is said to arise from X by an elementary modication if X is obtained from
X by replacing some entry v with
MATH 601, AUTUMN 2008
The index notation with the summing convention
The index notation includes the following conventions:
(a) In each term (monomial) forming a given expression, any index (that is, a subscript
or superscript) may appear at most twice.
(
MATH 601, AUTUMN 2008
Additional homework IV,
October 18
PROBLEMS
1. For any integer n 0 , let V be the vector space of all polynomials f of the
real variable x with deg f n . Determine if the linear operator T : V V , given by
(T f )(x) = f (x) + f (x 1)
MATH 601, AUTUMN 2007
Additional homework II,
October 2
PROBLEMS
1. Given a linear operator T : V W between nite-dimensional vector spaces
V, W , along with bases ej in V and ea in W , we characterize the components Tja of T
relative to the ej and ea by T
MATH 601, AUTUMN 2008
Additional homework I,
October 2
PROBLEMS
1. Determine if the set X consisting of the following three functions
t 1,
t2 + 3 ,
t t2
of the real variable t , spans the whole real vector space V of real-valued polynomials in t
of degree
MATH 601, AUTUMN 2007
Additional homework I,
October 2
PROBLEMS
1. Determine if the set X consisting of the following three functions
t 1,
t2 + 3 ,
t t2
of the real variable t , spans the whole real vector space V of real-valued polynomials in t
of degree
call # 13420-2
MATH 601, M-W-F 9:30 a.m., SO 0056
DETAILED LIST OF TOPICS
Sept. 22: VECTOR SPACES (real & complex), spaces of arrow vectors, the numerical
spaces IKn, function spaces, subspaces, nite sets of vectors as opposed to unordered
systems and ord