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Chapter 4: Q29E (page 191)

Write the difference equations in Exercises 29 and 30 as first order systems, \({x_{k + {\bf{1}}}} = A{x_k}\), for all \(k\).

\({y_{k + {\bf{4}}}} - {\bf{6}}{y_{k + {\bf{3}}}} + {\bf{8}}{y_{k + {\bf{2}}}} + {\bf{6}}{y_{k + {\bf{1}}}} - {\bf{9}}{y_k} = {\bf{0}}\)

Short Answer

Expert verified

\(A = \left[ {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\9&{ - 6}&{ - 8}&6\end{array}} \right]\)

Step by step solution

01

Write vectors \({x_k}\)and \({x_{k + {\bf{1}}}}\)

Vectors\({x_k}\)and\({x_{k + 1}}\)can be expressed as

\({x_k} = \left[ {\begin{array}{*{20}{c}}{{y_k}}\\{{y_{k + 1}}}\\{{y_{k + 2}}}\\{{y_{k + 3}}}\end{array}} \right]\) and \({x_{k + 1}} = \left[ {\begin{array}{*{20}{c}}{{y_{k + 1}}}\\{{y_{k + 2}}}\\{{y_{k + 3}}}\\{{y_{k + 4}}}\end{array}} \right]\).

02

Write \({x_{k + {\bf{1}}}}\) in the matrix form

Thematrix formfor \({x_{k + 1}}\) is

\(\begin{aligned} {x_{k + 1}} &= \left[ {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\9&{ - 6}&{ - 8}&6\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{y_k}}\\{{y_{k + 1}}}\\{{y_{k + 2}}}\\{{y_{k + 3}}}\end{array}} \right]\\ &= A{x_k}\end{aligned}\).

So, matrix \(A = \left[ {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\9&{ - 6}&{ - 8}&6\end{array}} \right]\).

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Most popular questions from this chapter

Let V be a vector space that contains a linearly independent set \(\left\{ {{u_{\bf{1}}},{u_{\bf{2}}},{u_{\bf{3}}},{u_{\bf{4}}}} \right\}\). Describe how to construct a set of vectors \(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\) in V such that \(\left\{ {{v_{\bf{1}}},{v_{\bf{3}}}} \right\}\) is a basis for Span\(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\).

Let \(V\) and \(W\) be vector spaces, and let \(T:V \to W\) be a linear transformation. Given a subspace \(U\) of \(V\), let \(T\left( U \right)\) denote the set of all images of the form \(T\left( {\mathop{\rm x}\nolimits} \right)\), where x is in \(U\). Show that \(T\left( U \right)\) is a subspace of \(W\).

Consider the polynomials \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + {t^{\bf{2}}},{p_{\bf{2}}}\left( t \right) = {\bf{1}} - {t^{\bf{2}}}\). Is \(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}}} \right\}\) a linearly independent set in \({{\bf{P}}_{\bf{3}}}\)? Why or why not?

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{3}}}\)in the plane\(x + {\bf{2}}y + z = {\bf{0}}\). (Hint:Think of the equation as a “system” of homogeneous equations.)

Let be a linear transformation from a vector space \(V\) \(T:V \to W\)in to vector space \(W\). Prove that the range of T is a subspace of . (Hint: Typical elements of the range have the form \(T\left( {\mathop{\rm x}\nolimits} \right)\) and \(T\left( {\mathop{\rm w}\nolimits} \right)\) for some \({\mathop{\rm x}\nolimits} ,\,{\mathop{\rm w}\nolimits} \)in \(V\).)\(W\)
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