Algebra questions

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 I’m 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
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[0,1]. Let S be the subspace of C2[0,1] consisting of all vectors 𝑓(𝑥) such that !# (0) = 0 and !# (1) +
𝑓(1) = 0.
Define P4 as the vector space of all polynomials of degree 4 or less. Two arbitrary elements of P4 have
representations as
𝑝(𝑥) = 𝑎$ + 𝑎% 𝑥 + 𝑎& 𝑥 & + 𝑎’ 𝑥 ‘ + 𝑎( 𝑥 (
𝑞(𝑥) = 𝑏$ + 𝑏% 𝑥 + 𝑏& 𝑥 & + 𝑏’ 𝑥 ‘ + 𝑏( 𝑥 (
Define an inner product on P4 by
%
(𝑝(𝑥), 𝑞(𝑥)) = . 𝑝(𝑥)𝑞(𝑥)𝑑𝑥
(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 ℎ(𝑥) = 𝑥 ( − 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,
𝑑
𝑑Θ
− 4𝛼(𝑥) 7 + 2𝛽(𝑥)Θ = 𝑔(𝑥)
𝑑𝑥
𝑑𝑥
with
Θ(0) = 0
𝑑Θ
(1) + Θ(1) = 0
𝑑𝑥
The problem is one of a differential operator defined such that
𝐿Θ = 𝑔(𝑥)
is defined on S. Show that 𝐿 is self adjoint on S with respect to the inner product
%
(𝑓, 𝑔) = . 𝑓(𝑥)𝑔(𝑥)𝑑𝑥
(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 £ x £ 1.