Question

(M) Let V be the space \(C\left( {0,2\pi } \right)\)with the inner product of Example 7. Use the Gram–Schmidt process to create an orthogonal basis for the subspace spanned by \(\left\{ {1,\cos t,{{\cos }^2}t,{{\cos }^3}t} \right\}\). Use a matrix program or computational program to compute the appropriate definite integrals.

Short answer

The orthogonal basis are \({f_0}\left( t \right) = 1\), \({f_1}\left( t \right) = \cos t\), \({f_2}\left( t \right) = \frac{1}{2}\cos 2t\) and \({f_3}\left( t \right) = \frac{1}{4}\cos 3t\) .

Step-by-step solution

Find the basis

It is given that \(V\) be the space \(C\left( {0,2\pi } \right)\) and an orthogonal basis for the subspace is spanned by \(\left\{ {1,\cos t,{{\cos }^2}t,{{\cos }^3}t} \right\}\).

Let the matrix be defined as \(A = \left( {\begin{array}{*{20}{c}}1\\{\cos t}\\{{{\cos }^2}t}\\{{{\cos }^3}t}\end{array}} \right)\).Use the following steps to find the associated values for the obtained data in MATLAB.

Formulate the matrix A using the commands as:

>>A=(1,cos(t),(cos(t))^2,(cos(t))^3);
>>(m,n) = size(A);

And to find\(QR\)by using the following commands:

>>fori = 1:n
>>v = A(:,i);
>>for j=1:i-1
>>R(j,i) = Q(:,j)'*A(:,i);
>>v = v - R(i,j)*Q(:,j);
>>end
>>R(i,i) = norm(v);
>>Q(:,i) = v/R(i,i);
>>end

By using the matrix program, the new orthogonal polynomials, are obtained as \({f_0}\left( t \right) = 1\), \({f_1}\left( t \right) = \cos t\), \({f_2}\left( t \right) = \frac{1}{2} - {\cos ^2}t\) and \({f_3}\left( t \right) = \frac{3}{4}\cos t - {\cos ^3}t\) .

Simplify the polynomials

By using the trigonometric identities, the simplified orthogonal basis is \({f_0}\left( t \right) = 1\), \({f_1}\left( t \right) = \cos t\), \({f_2}\left( t \right) = \frac{1}{2}\cos 2t\) and \({f_3}\left( t \right) = \frac{1}{4}\cos 3t\).