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Q7E

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Linear Algebra and its Applications
Found in: Page 191
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

In Exercise 7, find the coordinate vector \({\left( x \right)_{\rm B}}\) of x relative to the given basis \({\rm B} = \left\{ {{b_{\bf{1}}},...,{b_n}} \right\}\).

7. \({b_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\\{ - {\bf{3}}}\end{array}} \right),{b_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{\bf{9}}\end{array}} \right),{b_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{2}}}\\{\bf{4}}\end{array}} \right),x = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{9}}}\\{\bf{6}}\end{array}} \right)\)

Coordinate vector \({\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\3\end{array}} \right)\)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the system

Note that \({\left( x \right)_{\rm B}}\)be the solution of the system.

\(\begin{array}{c}\left( {\begin{array}{*{20}{c}}{{b_1}}&{{b_2}}\end{array}} \right){\left( x \right)_{\rm B}} = x\\\left( {\begin{array}{*{20}{c}}1&{ - 3}&2\\{ - 1}&4&{ - 2}\\{ - 3}&9&4\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{c_1}}\\{{c_2}}\\{{c_3}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}8\\{ - 9}\\6\end{array}} \right)\end{array}\)

Step 2: Find the reduced row echelon form

Its augmented matrix is \(\left( {\begin{array}{*{20}{c}}{{b_1}}&{{b_2}}&x\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&{ - 3}&2&8\\{ - 1}&4&{ - 2}&{ - 9}\\{ - 3}&9&4&6\end{array}} \right)\).

At row 2, add row 1 to row 2, i.e., \({R_2} \to {R_2} + {R_1}\).

Also, at row 3, multiply row 1 by 3 and add it to row 3, i.e., \({R_3} \to {R_3} + 3{R_1}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - 3}&2&8\\0&1&0&{ - 1}\\0&0&{10}&{30}\end{array}} \right)\)

At row 3, divide row 3 by 10.

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - 3}&2&8\\0&1&0&{ - 1}\\0&0&1&3\end{array}} \right)\)

At row 1, multiply row 2 by 3 and add it to row 1, i.e., \({R_1} \to {R_1} + 2{R_2}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&0&2&5\\0&1&0&{ - 1}\\0&0&1&3\end{array}} \right)\)

At row 1, multiply row 3 by 2 and subtract it from row 1, i.e., \({R_1} \to {R_1} - 2{R_3}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1}\\0&1&0&{ - 1}\\0&0&1&3\end{array}} \right)\)

This implies \({c_1} = - 1,{c_2} = - 1,\) and \({c_3} = 3\).

Step 3: Draw a conclusion

\(\begin{array}{c}{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{{c_1}}\\{{c_2}}\\{{c_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\3\end{array}} \right)\end{array}\)

Most popular questions for Math Textbooks

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work

\({\left( {{\bf{1}} - t} \right)^{\bf{2}}}\), \(t - {\bf{2}}{t^{\bf{2}}} + {t^{\bf{3}}}\), \({\left( {{\bf{1}} - t} \right)^{\bf{3}}}\)

In Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)

If A is a \({\bf{6}} \times {\bf{4}}\) matrix, what is the smallest possible dimension of Null A?

Suppose \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), \({{\bf{p}}_{\bf{3}}}\), and \({{\bf{p}}_{\bf{4}}}\) are specific polynomials that span a two-dimensional subspace H of \({P_{\bf{5}}}\). Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Given vectors, \({u_{\bf{1}}}\),….,\({u_p}\) and w in V, show that w is a linear combination of \({u_{\bf{1}}}\),….,\({u_p}\) if and only if \({\left( w \right)_B}\) is a linear combination of vectors \({\left( {{{\bf{u}}_{\bf{1}}}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).
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