## Solve a Linear Recurrence Relation Using Vector Space Technique

## Problem 321

Let $V$ be a real vector space of all real sequences

\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\]
Let $U$ be a subspace of $V$ defined by

\[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\]
Let $T$ be the linear transformation from $U$ to $U$ defined by

\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]

**(a)** Find the eigenvalues and eigenvectors of the linear transformation $T$.

**(b)** Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.