The Timpson Lyceum · Linear Algebra (2026)

4.8 Exercises

Problem sheet · 21 points · 3 extra credit

Applications to Difference Equations

1. 1. (3 pts) Verify that the signals $y_k = 2^k$ and $y_k = (-3)^k$ are solutions of the difference equation $y_{k+2} + y_{k+1} - 6y_k = 0$, and write the general solution.

2. 2. (3 pts) Find the general solution of $y_{k+2} - 5y_{k+1} + 6y_k = 0$.

3. 3. (3 pts) The difference equation $y_{k+2} - 4y_{k+1} + 4y_k = 0$ has the repeated auxiliary root $r = 2$; its general solution is $y_k = (c_1 + c_2 k)2^k$. Verify directly that both $y_k = 2^k$ and $y_k = k\,2^k$ satisfy the equation. (You need only check the two given signals — the repeated-root formula itself is beyond this section.)

4. 4. (3 pts) Solve the initial-value problem $y_{k+2} - 5y_{k+1} + 6y_k = 0$ with $y_0 = 1$ and $y_1 = 0$.

5. 5. (3 pts) Consider $y_{k+2} - 5y_{k+1} + 6y_k = 0$ with $\mathbf{x}_k = \begin{bmatrix} y_k \\ y_{k+1} \end{bmatrix}$.

   (a) Rewrite the equation as a first-order vector recurrence $\mathbf{x}_{k+1} = A\mathbf{x}_k$, and give the companion matrix $A$.

   (b) Its auxiliary equation is $r^2 - 5r + 6 = 0$; find its roots and write the general solution $y_k$.

   (c) For a root $r$, show that $\mathbf{x}_k = r^k\mathbf{v}$ solves $\mathbf{x}_{k+1} = A\mathbf{x}_k$ exactly when $(A - rI)\mathbf{v} = \mathbf{0}$; solve this system for $r = 2$ to find such a $\mathbf{v}$.

6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The set of all solutions of $y_{k+2} + a\,y_{k+1} + b\,y_k = 0$ (with $b \ne 0$) is a $2$-dimensional vector space.

   (b) The signals $y_k = 1$ (constant) and $y_k = 2^k$ are linearly independent.

   (c) If two signals solve a homogeneous linear difference equation, then so does their sum.

   (d) A single nonzero signal is always a basis for the solution space of a second-order homogeneous equation.

7. 7. (3 pts) Consider the $n$-th order homogeneous equation $y_{k+n} + a_{n-1}y_{k+n-1} + \cdots + a_0 y_k = 0$ (with $a_0 \ne 0$). Prove that its solution set is a subspace of the space $\mathbb{S}$ of all signals, and state its dimension.

8. 8. (3 pts ••) The Fibonacci recurrence is $y_{k+2} = y_{k+1} + y_k$. Find its general solution, then derive the closed form (Binet's formula) for the sequence with $y_0 = 0$ and $y_1 = 1$.