## Homework Statement

Apply the Gram-Schmidt orthogonalization procedure to the canonical basis $1, x, x^2, x^3, x^4$ in order to find an orthonormal basis for the space P4([0, 1]) with respect to the inner product <p(x), q(x)> =int(0,1) p(x)q(x) dx

AND USE THIS BASIS TO FIND THE ADJOINT OF THE LINEAR MAP! $A(ax^4 + bx^3 +cx^2 + dx + e) = cx^2$ (=cx^2 ... no cx^3... latex acts funny?)

## The Attempt at a Solution

Finding the orthogonal basis using the Gram-Schmit algorithm is slimply plugging numbers into a formula, so that is straight forward.

My Basis is:

$e1 =1$

$e2 = x-1/2$

$e3 = x^2+1/6-x$

$e4 = x^3-1/20+3/5*x-3/2*x^2$

$e5 = x^4+1/70-2/7*x+9/7*x^2-2*x^3$

Could somebody please guide me as to how I would

USE THIS BASIS TO FIND THE ADJOINT OF THE LINEAR MAP $A(ax^4 + bx^3 + cx^2 + dx + e) = cx^2$ (=cx^2 ... no bx^2... latex acts funny?)

I honestly have no idea where to start?

thought a good place to start would be to find the map of e_i with respect to A,

A(e1) = 0
A(e2) = 0
A(e3) = x^2
A(e4) = 3/2*x^2
A(e5) = 9/7*x^2

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