Chapter 1: Q29E (page 1)
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Suppose that such a system happens to be consistent. Explain why there must be an infinite number of solutions.
There is at least one free unknown that occurs.
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Solve the systems in Exercises 11‑14.
12.\(\begin{aligned}{c}{x_1} - 3{x_2} + 4{x_3} = - 4\\3{x_1} - 7{x_2} + 7{x_3} = - 8\\ - 4{x_1} + 6{x_2} - {x_3} = 7\end{aligned}\)
Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.
28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)
Let \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\0\\{ - 2}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\8\end{array}} \right],\) and \({\rm{y = }}\left[ {\begin{array}{*{20}{c}}h\\{ - 5}\\{ - 3}\end{array}} \right]\). For what values(s) of \(h\) is \(y\) in the plane generated by \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\)
Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vectorand two eigen vectorsof A (with eigen values respectively). For the given values of, draw a rough trajectory. Consider the future and the past of the system.
If Ais a matrix with eigenvalues 3 and 4 and if localid="1668109698541" is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" cannot exceed 4.
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