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Q24E

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Linear Algebra and its Applications
Found in: Page 331
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Use the Gram–Schmidt process as in Example 2 to produce an orthogonal basis for the column space of

\(A = \left( {\begin{aligned}{{}{r}}{ - 10}&{13}&7&{ - 11}\\2&1&{ - 5}&3\\{ - 6}&3&{13}&{ - 3}\\{16}&{ - 16}&{ - 2}&5\\2&1&{ - 5}&{ - 7}\end{aligned}} \right)\)

The orthogonal basis is,\(W = \left\{ {\left( {\begin{aligned}{{}{r}}{ - 10}\\2\\{ - 6}\\{16}\\2\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}3\\3\\{ - 3}\\0\\3\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}6\\0\\6\\6\\0\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}0\\5\\0\\0\\{ - 5}\end{aligned}} \right)} \right\}\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: \(QR\) factorization of a Matrix

A matrix with order \(m \times n\) can be written as the multiplication of an upper triangular matrix \(R\) and a matrix \(Q\) which is formed by applying the Gram–Schmidt orthogonalization process to the \({\rm{col}}\left( A \right)\).

The matrix \(R\) can be found by the formula \({Q^T}A = R\).

By applying Gram-Schmidt orthogonal process, we can determine the orthogonal basis for the column space of \(A\)

Step 2: Finding the matrix \(R\)

Given that, \(A = \left( {\begin{aligned}{{}{r}}{ - 10}&{13}&7&{ - 11}\\2&1&{ - 5}&3\\{ - 6}&3&{13}&{ - 3}\\{16}&{ - 16}&{ - 2}&5\\2&1&{ - 5}&{ - 7}\end{aligned}} \right)\).

Now with the help of MATLAB, we shall find the orthogonal basis of the column space

MATLAB Command:

Enter matrix A in MATLAB.

>> A=(-10 13 7 -11; 2 1 5 3; -6 3 13 -3; 16 -16 -2 5; 2 1 -5 -7);

The required function:

function(B) = GramSchmidt(A)

(m,n) = size(A);

(U, jb) = rref(A);

x = length(jb);

B = zeros(m,x);

for i = 1:x

C(:,i)= A(:,(jb(i)));

end

B=C;

for i = 2:x

for j = 1:i-1

B(:,i) = C(:,i)- dot(C(:,i),B(:,j))/dot(B(:,j),B(:,j))* B(:,j) ;

end

end

end

Find the orthogonal basis:

(B) = GramSchmidt(A)

\(\begin{aligned}{}B &= \\\begin{aligned}{{}{r}}{ - 10.0000}&{3.0000}&{8.5000}&{ - 7.9620}\\{2.0000}&{3.0000}&{6.5000}&{5.3232}\\{ - 6.0000}&{ - 3.0000}&{11.5000}&{1.1103}\\{16.0000}&0&{ - 2.0000}&{4.2852}\\{2.0000}&{3.0000}&{ - 3.5000}&{ - 8.2510}\end{aligned}\end{aligned}\)

So, the orthogonal basis is\(W = \left\{ {\left( {\begin{aligned}{{}{r}}{ - 10}\\2\\{ - 6}\\{16}\\2\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}3\\3\\{ - 3}\\0\\3\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}6\\0\\6\\6\\0\end{aligned}} \right),\left( {\begin{aligned}{{}{r}}0\\5\\0\\0\\{ - 5}\end{aligned}} \right)} \right\}\).

Most popular questions for Math Textbooks

In Exercises 1-4, find a least-sqaures solution of \(A{\bf{x}} = {\bf{b}}\) by (a) constructing a normal equations for \({\bf{\hat x}}\) and (b) solving for \({\bf{\hat x}}\).

3. \(A = \left( {\begin{aligned}{{}{}}{\bf{1}}&{ - {\bf{2}}}\\{ - {\bf{1}}}&{\bf{2}}\\{\bf{0}}&{\bf{3}}\\{\bf{2}}&{\bf{5}}\end{aligned}} \right)\), \({\bf{b}} = \left( {\begin{aligned}{{}{}}{\bf{3}}\\{\bf{1}}\\{ - {\bf{4}}}\\{\bf{2}}\end{aligned}} \right)\)

To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from \(t = 0\) to \(t = 12\). The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7, 159.1, 222.0, 294.5, 380.4, 471.1, 571.7, 686.8, 809.2.

a. Find the least-squares cubic curve \(y = {\beta _0} + {\beta _1}t + {\beta _2}{t^2} + {\beta _3}{t^3}\) for these data.

b. Use the result of part (a) to estimate the velocity of the plane when \(t = 4.5\) seconds.

In Exercises 13 and 14, find the best approximation to \[{\bf{z}}\] by vectors of the form \[{c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2}\].

13. \[z = \left[ {\begin{aligned}3\\{ - 7}\\2\\3\end{aligned}} \right]\], \[{{\bf{v}}_1} = \left[ {\begin{aligned}2\\{ - 1}\\{ - 3}\\1\end{aligned}} \right]\], \[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\1\\0\\{ - 1}\end{aligned}} \right]\]

Find an orthogonal basis for the column space of each matrix in Exercises 9-12.

11. \(\left( {\begin{aligned}{{}{}}1&2&5\\{ - 1}&1&{ - 4}\\{ - 1}&4&{ - 3}\\1&{ - 4}&7\\1&2&1\end{aligned}} \right)\)

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 \(R = {Q^T}A\).
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