91Ó°ÊÓ

When the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm x}\nolimits} _3}{{\mathop{\rm v}\nolimits} _3} = {\mathop{\rm u}\nolimits} \) isconsistent, vector u is in thesubspace generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

The augmented matrix of the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm x}\nolimits} _3}{{\mathop{\rm v}\nolimits} _3} = {\mathop{\rm u}\nolimits} \) is shown below:

\(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}&{\mathop{\rm u}\nolimits} \end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\{ - 2}&{ - 7}&{ - 8}&{10}\\4&9&6&{ - 7}\\3&7&5&{ - 5}\end{array}} \right]\)

The following row operation demonstrates that u is not in the subspace generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

At row two, multiply row one by 2 and add it to row two. At row three, multiply row one by 4 and subtract it from row three. At row four, multiply row one by 3 and subtract it from row four.

\( \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\0&1&2&2\\0&{ - 7}&{ - 14}&9\\0&{ - 5}&{ - 10}&7\end{array}} \right]\)

At row three, multiply row two by 7 and add it to row three. At row four, multiply row two by 5 and add it to row four.

\( \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\0&1&2&2\\0&0&0&{23}\\0&0&0&{17}\end{array}} \right]\)

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

.

The augmented matrix represents an inconsistent system. Therefore, \({\mathop{\rm u}\nolimits} \) is not in the subspace of \({\mathbb{R}^4}\) generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).