Question

Let \({{\bf{u}}_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{4}}\\{ - {\bf{6}}}\end{array}} \right]\), \({{\bf{u}}_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{{b_{\bf{1}}}}\\{{b_{\bf{2}}}}\\{{b_{\bf{3}}}}\end{array}} \right]\), and \(W = {\bf{Span}}\left\{ {{{\bf{u}}_{\bf{1}}},\,{{\bf{u}}_{\bf{2}}}} \right\}\). Find an implicit description of W, that is, find a set of one or more homogenous equations that characterize the points of W.

Short answer

W is the set of points that satisfy \({b_1} + 2{b_2} + {b_3} = 0\).

Step-by-step solution

 Step 1:Write the system of equation

Consider the equation \({x_1}{{\bf{u}}_1} + {x_2}{{\bf{u}}_2} = {\bf{b}}\).

Then, the system of equation can be given in the following manner:

\({x_1}\left[ {\begin{array}{*{20}{c}}{ - 2}\\4\\{ - 6}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}1\\2\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{b_1}}\\{{b_2}}\\{{b_3}}\end{array}} \right]\)

The augmented matrix form is shown below:

\(\left[ {\begin{array}{*{20}{c}}{ - 2}&1&{{b_1}}\\4&2&{{b_2}}\\{ - 6}&{ - 5}&{{b_3}}\end{array}} \right]\)

Write the row reduce form for augmented matrix

Apply row operations to the aforementioned augmented matrixin the following manner:

\(\begin{aligned}{}A &= \left[ {\begin{aligned}{{}}{ - 2}&1&{{b_1}}\\4&2&{{b_2}}\\{ - 6}&{ - 5}&{{b_3}}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}}{ - 2}&1&{{b_1}}\\0&4&{{b_2} + 2{b_1}}\\0&{ - 8}&{{b_3} - 3{b_1}}\end{aligned}} \right]\,\,\,\,\,\,\,\left\{ {{R_2} \to {R_2} + 2{R_1},\,\,{R_3} \to {R_3} - 3{R_1}} \right\}\\ &= \left[ {\begin{aligned}{{}}{ - 2}&1&{{b_1}}\\0&4&{{b_2} + 2{b_1}}\\0&0&{{b_1} + 2{b_3} + {b_3}}\end{aligned}} \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to {R_3} + 2{R_2}} \right\}\end{aligned}\)

If the system has a consistent solution, then all elements of the last row of A must be equal to zero.

\({b_1} + 2{b_2} + {b_3} = 0\)

Thus, W is the set of points that satisfy \({b_1} + 2{b_2} + {b_3} = 0\).