Question

In Exercies 7-12, assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems.

\({\left( { - {\bf{1}}} \right)^k}\), \(k{\left( { - {\bf{1}}} \right)^k}\), \({{\bf{5}}^k}\), \({y_{k + {\bf{3}}}} - {\bf{3}}{y_{k + {\bf{2}}}} - {\bf{9}}{y_{k + {\bf{1}}}} - {\bf{5}}{y_k} = {\bf{0}}\)

Short answer

The dimension of H is 3, so the three linearly independent signals form a basis of H.

Step-by-step solution

Write the Casorati matrix

The Casorati matrix of the solution is

\({A_k} = \left[ {\begin{array}{*{20}{c}}{{{\left( { - 1} \right)}^k}}&{k{{\left( { - 1} \right)}^k}}&{{5^k}}\\{{{\left( { - 1} \right)}^{k + 1}}}&{\left( {k + 1} \right){{\left( { - 1} \right)}^{k + 1}}}&{{5^{k + 1}}}\\{{{\left( { - 1} \right)}^{k + 2}}}&{\left( {k + 2} \right){{\left( { - 1} \right)}^{k + 2}}}&{{5^{k + 2}}}\end{array}} \right]\).

Check the Casorati matrix for \(k = {\bf{0}}\)

Substitute 0 for k in the Casorati matrix.

\(\begin{aligned} {A_0} &= \left[ {\begin{array}{*{20}{c}}{{{\left( { - 1} \right)}^0}}&{0{{\left( { - 1} \right)}^k}}&{{5^0}}\\{{{\left( { - 1} \right)}^{0 + 1}}}&{\left( {0 + 1} \right){{\left( { - 1} \right)}^{0 + 1}}}&{{5^{0 + 1}}}\\{{{\left( { - 1} \right)}^{0 + 2}}}&{\left( {0 + 2} \right){{\left( { - 1} \right)}^{0 + 2}}}&{{5^{0 + 2}}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}1&0&1\\1&{ - 1}&1\\1&2&{25}\end{array}} \right]\\ &\sim \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\end{aligned}\)

The Casorati matrix is row equivalent to the identity matrix. Therefore, it is invertible.

Hence, the set of signals \(\left\{ {{{\left( { - 1} \right)}^k},\,k{{\left( { - 1} \right)}^k},\;{5^k}} \right\}\) islinearly independent. The dimension of H is 3, so the three linearly independent signals form a basis of H.