CAUCHY-BINET
JANUARY 6, 2009
Theorem 0.1. (Cauchy-Binet) Let A be a k n matrix and B be an n k
matrix. Then
det(AB) =
det(A(J) det(B(J),
J
where J = (j1 , j2 , . . . , jk ), 1 j1 < j2 < < jk n, runs through all such
multi-indices, A(J) denotes the matrix
Change of Variable Formula for Multiple Integrals
S. Kumaresan and G. Santhanam
Let f : [a, b] R be a continuous function and : [c, d] [a, b] be a C 1 -function. Then
the change of variable formula for one variable integrals says that
t
(t)
f (x) (x)dx =
21-236 Analysis Assignment 1
Problems due Wednesday, January 29
1.1. Pugh p197 #58 (log-harmonic series)
1.2. Let a = (ak ) and b = (bk ) be sequences of real numbers, and let
k=0
k=0
their correpsonding generating functions be given by the power series
a
21-236 Analysis Assignment 3
Problems due Wednesday, February 26
3.1. (cf. Pugh p 279 Theorem 14) Assume f : [a, b] Y Z is continuous,
where Y and Z are Banach spaces and [a, b] R. Dene F : Y Z by the
Riemann integral
b
F (y) =
f (x, y) dx.
a
Assume the (
Proof of the Cauchy-Binet determinant formula
Let 1 k n and suppose A and B are nk matrices. We introduce the following notation
for the n possible k k submatrices of A and B: Let In,k denote the set of k-element
k
subsets of [n] := cfw_1, . . . , n. For
21-236 Analysis Assignment 4 Problems due Wednesday, April 9
4.1. Let x, y : [0, 1] R be C 1 . Use Fubinis theorem and the CauchySchwarz inequality for the inner product of (|x (t)|, |y (t)|) with (|x (s)|, |y (s)|)
to prove Chams inequality (named after
21-236 Analysis Assignment 2
Problems due Wednesday, February 12
2.1. Suppose f : C C is entire, meaning complex dierentiable at every
point in C. Suppose further that
f (z)
0
z
as |z| .
Prove that f is constant; I.e., prove that for every z1 , z2 C, f (z
21-236 Analysis Assignment 5
Problems due Wednesday, April 30
5.1. (a) Suppose = I[n]k cI dxI is a k-form on an open set U Rn ,
with constant coecient functions cI . Show is exact.
(b) Let be a k-form and an -form on Rn . If is closed and is exact,
show i