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.