91Ó°ÊÓ

Given \(A\) and \(b\) in Exercises 11 and 12, write the augmented matrix for the linear system that corresponds to the matrix equation \(Ax = b\). Then solve the system and write the solution as a vector.

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

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

11.\({a_1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\0\end{array}} \right],{a_2} = \left[ {\begin{array}{*{20}{c}}0\\1\\2\end{array}} \right],{a_3} = \left[ {\begin{array}{*{20}{c}}5\\{ - 6}\\8\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\)

Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer.

\(\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\4\end{array}} \right],\left[ {\begin{array}{*{20}{c}}3\\{ - 5}\\7\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 1}\\5\\h\end{array}} \right]\)

In Exercises 7-12, describe all solutions of \(Ax = 0\) in parametric vector form, where \(A\) is row equivalent to the given matrix.

12. \(\left( {\begin{array}{*{20}{c}}1&5&2&{ - 6}&9&0\\0&0&1&{ - 7}&4&{ - 8}\\0&0&0&0&0&1\\0&0&0&0&0&0\end{array}} \right)\)

Given \(A\) and \(b\) in Exercises 11 and 12, write the augmented matrix for the linear system that corresponds to the matrix equation \(Ax = b\). Then solve the system and write the solution as a vector.

12. \({\mathop{\rm A}\nolimits} = \left[ {\begin{array}{*{20}{c}}1&2&1\\{ - 3}&{ - 1}&2\\0&5&3\end{array}} \right],b = \left[ {\begin{array}{*{20}{c}}0\\1\\{ - 1}\end{array}} \right]\)

a. Find the general traffic pattern in the freeway network shown in the figure.(Flow rates are in cars/minute).

b. Describe the general traffic pattern when the road whose flow is \({x_4}\)is closed.

c. When \({x_4} = 0\), what is minimum value of \({x_1}\)?

Find the general solutions of the systems whose augmented matrices are given as

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation such that \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) are the vectors shown in the figure. Using the figure, sketch the vector \(T\left( {2,1} \right)\).

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

a. Find the general flow pattern in the network shown in the figure.

b. Assuming that the flow must be in the directions indicated, find the minimum flows in the branches denoted by \({x_2}\), \({x_3}\), \({x_4}\) and \({x_5}\).