91Ó°ÊÓ

When the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} = {\mathop{\rm w}\nolimits} \) isconsistent, vector w is in thesubspace generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

The augmented matrix for the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} = {\mathop{\rm w}\nolimits} \) is shown below:

\(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{\mathop{\rm w}\nolimits} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2&{ - 4}&8\\3&{ - 5}&2\\{ - 5}&8&{ - 9}\end{array}} \right]\)

The following row operation demonstrates that w is not in the subspace generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

At row one, multiply row one by \(\frac{1}{2}\).

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\3&{ - 5}&2\\{ - 5}&8&{ - 9}\end{array}} \right]\)

At row two, multiply row one by \(3\) and subtract it from row two. At row three, multiply row one by 5 and add it to row three.

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\0&1&{ - 10}\\0&{ - 2}&{11}\end{array}} \right]\)

At row three, multiply row two by 2 and add it to row three.

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\0&1&{ - 10}\\0&0&{ - 9}\end{array}} \right]\)

Mark the pivot column in the row echelon form of the matrix

.

The augmented matrix represents an inconsistent system. Therefore, w is not in the subspace of \({\mathbb{R}^3}\) generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).