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Q37E

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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

(M) Determine whether w is in the column space of \(A\), the null space of \(A\), or both, where

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\1\\{ - 1}\\{ - 3}\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1\\{ - 5}&{ - 1}&0&{ - 2}\\9&{ - 11}&7&{ - 3}\\{19}&{ - 9}&7&1\end{array}} \right)\)

\({\mathop{\rm w}\nolimits} \) is in Col A, and w is not in Nul A.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write an augmented matrix

Consider the augmented matrix \(\left( {\begin{array}{*{20}{c}}A&{\mathop{\rm w}\nolimits} \end{array}} \right)\) shown below:

\(\left( {\begin{array}{*{20}{c}}A&w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right)\)

Step 2: Convert the matrix into row-reduced echelon form

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right)\).

Use the following code in MATLAB to obtain the row-reduced echelon form.

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {7\,\,\,6\,\,\, - 4\,\,\,1\,\,\,1;\,\, - 5\,\,\, - 1\,\,\,0\,\,\, - 2\,\,\,1;\,\,9\,\,\, - 11\,\,\,7\,\,\, - 3\,\,\, - 1;\,19\,\,\, - 9\,\,\,7\,\,\,1\,\,\, - 3} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\(\left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&0&{\frac{{ - 1}}{{95}}}&{\frac{1}{{95}}}\\0&1&0&{\frac{{39}}{{19}}}&{\frac{{ - 20}}{{19}}}\\0&0&1&{\frac{{267}}{{95}}}&{\frac{{ - 172}}{{95}}}\\0&0&0&0&0\end{array}} \right)\)

Step 3: Determine whether w is in the column space of A

A typical vector v in Col A has the property that the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm v}\nolimits} \) is consistent.

The system of equations of matrix A is consistent.

Thus, \({\mathop{\rm w}\nolimits} \) is in Col A.

Step 4: Determine whether w is in the null space of A

A typical vector v in Nul A has the property that \(A{\mathop{\rm v}\nolimits} = 0\).

Use the code in MATLAB to compute the matrix \({\mathop{\rm Aw}\nolimits} \) as shown below:

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {7\,\,\,6\,\,\, - 4\,\,\,1;\,\, - 5\,\,\, - 1\,\,\,0\,\,\, - 2;\,\,9\,\,\, - 11\,\,\,7\,\,\, - 3;\,19\,\,\, - 9\,\,\,7\,\,\,1} \right)\\ > > {\mathop{\rm w}\nolimits} = \left( {1;\,\,1;\,\, - 1;\,\, - 3} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm A}\nolimits} * w\end{array}\)

\(\begin{array}{c}{\mathop{\rm A}\nolimits} {\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1\\{ - 5}&{ - 1}&0&{ - 2}\\9&{ - 11}&7&{ - 3}\\{19}&{ - 9}&7&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\1\\{ - 1}\\{ - 3}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{14}\\0\\0\\0\end{array}} \right)\end{array}\)

Thus, w is not in Nul A.

Most popular questions for Math Textbooks

Let be a basis of \({\mathbb{R}^n}\). .Produce a description of an \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)matrix A that implements the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\). Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)). (See Exercise 21.)

In Exercise 1, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

1. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{6}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{array}} \right)\)

Let \(B = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\end{array}} \right),\,\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{9}}\end{array}} \right)\,} \right\}\). Since the coordinate mapping determined by B is a linear transformation from \({\mathbb{R}^{\bf{2}}}\) into \({\mathbb{R}^{\bf{2}}}\), this mapping must be implemented by some \({\bf{2}} \times {\bf{2}}\) matrix A. Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)).

Consider the polynomials , and \({p_{\bf{3}}}\left( t \right) = {\bf{2}}\) \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + t,{p_{\bf{2}}}\left( t \right) = {\bf{1}} - t\)(for all t). By inspection, write a linear dependence relation among \({p_{\bf{1}}},{p_{\bf{2}}},\) and \({p_{\bf{3}}}\). Then find a basis for Span\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

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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