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Q38E

Expert-verifiedFound in: Page 191

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

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

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\)

\({\mathop{\rm w}\nolimits} \) is in Col *A,* and w is in Nul *A*.

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

\(\left( {\begin{array}{*{20}{c}}A&w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)\)

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)\).

Use the code in MATLAB to obtain the row-reduced echelon form as shown below:

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0\,\,\,1;\, - 5\,\,\,2\,\,\,1\,\,\, - 2\,\,\,2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3\,\,\,1;\,3\,\,\, - 2\,\,\,1\,\,\,\,0\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\(\left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&{ - 1}&0&{ - 2}\\0&1&{ - 2}&0&{ - 3}\\0&0&1&1&1\\0&0&0&0&0\end{array}} \right)\)

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 the equation of matrix *A* is consistent.

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

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

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

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0;\, - 5\,\,\,2\,\,\,1\,\,\, - 2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3;\,3\,\,\, - 2\,\,\,1\,\,\,\,0} \right)\\ > > {\mathop{\rm w}\nolimits} = \left( {1;\,\,2;\,\,1;\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm A}\nolimits} * w\end{array}\)

\(\begin{array}{c}{\mathop{\rm A}\nolimits} {\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}0\\0\\0\\0\end{array}} \right)\end{array}\)

Thus, w is in Nul *A*.

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