Question

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

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

Short answer

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

Step-by-step solution

Write an augmented matrix

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

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

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.

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

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

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 inCol A.

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.