Let T:
be defined as
. Prove or disprove that
can
be written as the sum of two one-dimensional, T-invariant
subspaces.
Let T: be defined as . Prove or disprove that can be written as the sum of two one-dimensional, T-invariant subspace...
1. Let and be subspaces of
. Prove
that is also a
subspace of .
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Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
Let
be a map
Define the map
prove or disprove
2)
for all
3)
for all
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Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
Let T: V
V and S: V
V and R: V
V be three linear operators on V. Suppose we have
T
S= S
R , Then prove ker(S) is an invariant subspace for R .
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Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
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a) Let
. Show that
.
b) Show that the derivative can be written as:
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Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
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Let be a topological space, let and be paths in such that . Show that defined by is a path in We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image