(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

Expert verified

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

01

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 \({f_3}\left( t \right) = \frac{3}{4}\cos t - {\cos ^3}t\) .

02

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 \({f_3}\left( t \right) = \frac{1}{4}\cos 3t\).

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Most popular questions from this chapter

Find the distance between \({\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{10}\\{ - 3}\end{aligned}} \right)\) and \({\mathop{\rm y}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\{ - 5}\end{aligned}} \right)\).

In Exercises 3–6, verify that\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\]is an orthogonal set, and then find the orthogonal projection of\[y\]onto Span\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\].

6.\[{\rm{y}} = \left[ {\begin{aligned}6\\4\\1\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}{ - 4}\\{ - 1}\\1\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}0\\1\\1\end{aligned}} \right]\]

In Exercises 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

6. \(\left( {\begin{aligned}{{}}3\\{ - 1}\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 5}\\9\\{ - 9}\\3\end{aligned}} \right)\)

In Exercises 17 and 18, all vectors and subspaces are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

17. a.If \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) is an orthogonal basis for\(W\), then multiplying

\({v_3}\)by a scalar \(c\) gives a new orthogonal basis \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},c{{\bf{v}}_3}} \right\}\).

b. The Gram–Schmidt process produces from a linearly independent

set \(\left\{ {{{\bf{x}}_1}, \ldots ,{{\bf{x}}_p}} \right\}\)an orthogonal set \(\left\{ {{{\bf{v}}_1}, \ldots ,{{\bf{v}}_p}} \right\}\) with the property that for each \(k\), the vectors \({{\bf{v}}_1}, \ldots ,{{\bf{v}}_k}\) span the same subspace as that spanned by \({{\bf{x}}_1}, \ldots ,{{\bf{x}}_k}\).

c. If \(A = QR\), where \(Q\) has orthonormal columns, then \(R = {Q^T}A\).

A Householder matrix, or an elementary reflector, has the form \(Q = I - 2{\bf{u}}{{\bf{u}}^T}\) where u is a unit vector. (See Exercise 13 in the Supplementary Exercise for Chapter 2.) Show that Q is an orthogonal matrix. (Elementary reflectors are often used in computer programs to produce a QR factorization of a matrix A. If A has linearly independent columns, then left-multiplication by a sequence of elementary reflectors can produce an upper triangular matrix.)

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