3. If A and B are finite dimensional, orthogonal subspaces of an inner product space V, then show dim(A O B) = dimA + dimB. If A and B are not necessarily orthogonal. then what is the relationship between dim(A+ B), dimA, and dim B? 4. (a) Let V, be the spaces generated hy

3. If A and B are finite dimensional, orthogonal subspaces of an inner product space V, then show dim(A O B) = dimA + dimB. If A and B are not necessarily orthogonal. then what is the relationship between dim(A+ B), dimA, and dim B? 4. (a) Let V, be the spaces generated hy

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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pert Boggess, Francis I. Narcowich-A First Course in.X +
3. 3/4 X<1.
3. If A and B are finite-dimensional, orthogonal subspaces of an inner product
space V, then show
dim(A e B) = dimA + dimB.
If A and B are not necessarily orthogonal. then what is the relationship
between dim(A + B), dimA, and dim B?
4. (a) Let V be the spaces generated by o(2" x– k), k e Z, where o is the
Haar scaling funetion. On the interval 0 <x < 1, what are the dimen-
sions of the spaces V, and W for n20 (just count the number of basis
elements)?
(b) Using the result of problem 3, count the dimension of the space on the
right side of the cquality
V.= W W, 2 © W, Vo.
Is your answer the same as the one you computed for dim V in part (a)?
5. Let ø and v be the Haar scaling and wavelet functions, respectively. Let V
and W, be the spaces generated by 0(2x- k), kEZ. and V (2x-k). KE
Z, respectively. Suppose f(x) = £ d0(2x = k) (a e R belongs 10 V
Show explicitly that if f is orthogonal to each basis element o(x DE V
for all integersI, then ay = -ay for all integers I and hence show
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