need help with these algebra questions. Please show work and answer in a word document. DO NOT TAKE THE QUESTION IF YOU CANT COMPLETE IT. Somebody already canceled and Iām running out of time plz help!
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$ + š% š„ + š& š„ & + š’ š„ ‘ + š( š„ (
š(š„) = š$ + š% š„ + š& š„ & + š’ š„ ‘ + š( š„ (
Define an inner product on P4 by
%
(š(š„), š(š„)) = . š(š„)š(š„)šš„
(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(š„) = š„ ( ā 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
šĪ
ā 4š¼(š„) 7 + 2š½(š„)Ī = š(š„)
šš„
šš„
with
Ī(0) = 0
šĪ
(1) + Ī(1) = 0
šš„
The problem is one of a differential operator defined such that
šæĪ = š(š„)
is defined on S. Show that šæ is self adjoint on S with respect to the inner product
%
(š, š) = . š(š„)š(š„)šš„
(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.