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.

\({{\bf{2}}^k}\), \({{\bf{4}}^k}\), \({\left( { - {\bf{5}}} \right)^k}\), \({y_{k + {\bf{3}}}} - {y_{k + {\bf{2}}}} - {\bf{22}}{y_{k + {\bf{1}}}} + {\bf{40}}{y_k} = {\bf{0}}\)

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

\({A_k} = \left[ {\begin{array}{*{20}{c}}{{2^k}}&{{4^k}}&{{{\left( { - 5} \right)}^k}}\\{{2^{k + 1}}}&{{4^{k + 1}}}&{{{\left( { - 5} \right)}^{k + 1}}}\\{{2^{k + 2}}}&{{4^{k + 2}}}&{{{\left( { - 5} \right)}^{k + 2}}}\end{array}} \right]\).

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

Substitute 0 for k in the Casorati matrix.

\(\begin{aligned}{A_0} &= \left[ {\begin{array}{*{20}{c}}{{2^0}}&{{4^0}}&{{{\left( { - 5} \right)}^0}}\\{{2^{0 + 1}}}&{{4^{0 + 1}}}&{{{\left( { - 5} \right)}^{0 + 1}}}\\{{2^{0 + 2}}}&{{4^{0 + 2}}}&{{{\left( { - 5} \right)}^{0 + 2}}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}1&1&1\\2&4&{ - 5}\\4&{16}&{25}\end{array}} \right]\\ &\sim \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\end{aligned}\)

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

Hence, the set of signals \(\left\{ {{1^k},\,{2^k},\;{{\left( { - 2} \right)}^k}} \right\}\) is linearly independent. The dimension of H is 3, so the three linearly independent signals form a basis of H.