Chapter 4: Q31E (page 191)
Is the following difference equation of order 3? Explain.\({y_{k + 3}} + 5{y_{k + 2}} + 6{y_{k + 1}} = 0\).
No, the order of the given difference equation is 2.
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Question: Determine if the matrix pairs in Exercises 19-22 are controllable.
21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).
Justify the following equalities:
a.\({\rm{dim Row }}A{\rm{ + dim Nul }}A = n{\rm{ }}\)
b.\({\rm{dim Col }}A{\rm{ + dim Nul }}{A^T} = m\)
Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\) if \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\) and \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]\).
In Exercise 2, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).
2. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{5}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{7}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{5}}}\end{array}} \right)\)
Explain what is wrong with the following discussion: Let \({\bf{f}}\left( t \right) = {\bf{3}} + t\) and \({\bf{g}}\left( t \right) = {\bf{3}}t + {t^{\bf{2}}}\), and note that \({\bf{g}}\left( t \right) = t{\bf{f}}\left( t \right)\). Then, \(\left\{ {{\bf{f}},{\bf{g}}} \right\}\) is linearly dependent because g is a multiple of f.
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