Question

In Exercises 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.

${\mathbf{1}}^k$, ${\mathbf{2}}^k$, ${\left(-\mathbf{2}\right)}^k$, $y_{k+\mathbf{3}}-y_{k+\mathbf{2}}-4y_{k+\mathbf{1}}+\mathbf{4}{y}_k=\mathbf{0}$

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

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

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

Substitute 0 for k in the Casorati matrix.

$\begin{array}{rl}{A}_0& =\left[\begin{array}{ccc}{1}^0& 2^0& {\left(-2\right)}^0\\ 1^{0+1}& 2^{0+1}& {\left(-2\right)}^{0+1}\\ 1^{0+2}& 2^{0+2}& {\left(-2\right)}^{0+2}\end{array}\right]\\ & =\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& -2\\ 1& 4& 4\end{array}\right]\\ & \sim \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$

The Casorati matrix is row equivalent to 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.