Algebra questions

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4600:660 Engineering Analysis
HW Set 2
Linear Algebra
Let C2[0,1] be the vector space consisting of all functions that are twice differentiable everywhere on
!”
!”
[0,1]. Let S be the subspace of C2[0,1] consisting of all vectors š‘“(š‘„) such that !# (0) = 0 and !# (1) +
š‘“(1) = 0.
Define P4 as the vector space of all polynomials of degree 4 or less. Two arbitrary elements of P4 have
representations as
š‘(š‘„) = š‘Ž$ + š‘Ž% š‘„ + š‘Ž& š‘„ & + š‘Ž’ š‘„ ‘ + š‘Ž( š‘„ (
š‘ž(š‘„) = š‘$ + š‘% š‘„ + š‘& š‘„ & + š‘’ š‘„ ‘ + š‘( š‘„ (
Define an inner product on P4 by
%
(š‘(š‘„), š‘ž(š‘„)) = . š‘(š‘„)š‘ž(š‘„)š‘‘š‘„
(1)
$
Let W be the intersection between S and P4.
1. Determine a basis for W.
2. Determine an orthonormal basis for W with respect to the inner product defined in Equation (1).
3. Expand the vector in W defined by ā„Ž(š‘„) = š‘„ ( āˆ’ 5 in terms of the orthonormal basis vectors for
W.
4. Now (for the remainder of the HW Set) , consider a non-dimensional heat transfer problem for
an extended surface,
š‘‘
š‘‘Ī˜
āˆ’ 4š›¼(š‘„) 7 + 2š›½(š‘„)Ī˜ = š‘”(š‘„)
š‘‘š‘„
š‘‘š‘„
with
Ī˜(0) = 0
š‘‘Ī˜
(1) + Ī˜(1) = 0
š‘‘š‘„
The problem is one of a differential operator defined such that
šæĪ˜ = š‘”(š‘„)
is defined on S. Show that šæ is self adjoint on S with respect to the inner product
%
(š‘“, š‘”) = . š‘“(š‘„)š‘”(š‘„)š‘‘š‘„
(2)
$
5. Show that L is positive definite on S with respect to the standard inner product, Equation (2), on
S. Assume that a ( x) > 0 and b ( x) > 0 for all x, 0 Ā£ x Ā£ 1.

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