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.

${\left(-\mathbf{1}\right)}^k$, $k{\left(-\mathbf{1}\right)}^k$, ${\mathbf{5}}^k$, $y_{k+\mathbf{3}}-\mathbf{3}{y}_{k+\mathbf{2}}-\mathbf{9}{y}_{k+\mathbf{1}}-\mathbf{5}{y}_k=\mathbf{0}$

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

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

Check the Casorati matrix for $k=\mathbf{0}$

Substitute 0 for k in the Casorati matrix.

$\begin{array}{rl}{A}_0& =\left[\begin{array}{ccc}{\left(-1\right)}^0& 0{\left(-1\right)}^k& 5^0\\ {\left(-1\right)}^{0+1}& \left(0+1\right){\left(-1\right)}^{0+1}& 5^{0+1}\\ {\left(-1\right)}^{0+2}& \left(0+2\right){\left(-1\right)}^{0+2}& 5^{0+2}\end{array}\right]\\ & =\left[\begin{array}{ccc}1& 0& 1\\ 1& -1& 1\\ 1& 2& 25\end{array}\right]\\ & \sim \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$

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.