Sequences Satisfying Linear Recurrence Relation Form a Subspace
Problem 308
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\]
Let $U$ be the subset of $V$ defined by
\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\]
Prove that $U$ is a subspace of $V$.
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Proof.
We prove the subspace criteria.
- The zero vector in $V$ is in $U$.
- For any two elements $(a_i)_{i=1}^{\infty}, (b_i)_{i=1}^{\infty}\in U$, we have $(a_i)_{i=1}^{\infty}+(b_i)_{i=1}^{\infty} \in U$.
- For any scalar $c$ and any element $(a_i)_{i=1}^{\infty} \in U$, we have $c(a_i)_{i=1}^{\infty} \in U$.
The zero vector in $V$ is the zero sequence $(0)=(0,0,0,\dots)$. Clearly, this sequence satisfies the recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$. Thus the zero vector $(0)\in U$, hence condition 1 is met.
To check condition 2, take $(a_i)_{i=1}^{\infty}, (b_i)_{i=1}^{\infty}\in U$.
We want to show that the sum
\[(a_i)_{i=1}^{\infty}+(b_i)_{i=1}^{\infty}=(a_i+b_i)_{i=1}^{\infty}\]
is also in $U$.
Since $(a_i)_{i=1}^{\infty}, (b_i)_{i=1}^{\infty}\in U$, these sequences satisfy the recurrence relation
\[a_{k+2}-5a_{k+1}+3a_{k}=0 \tag{*}\]
and
\[b_{k+2}-5b_{k+1}+3b_{k}=0\]
for $k=1, 2, \dots$.
Using these two relations, we have
\begin{align*}
&(a_{k+2}+b_{k+2})-5(a_{k+1}+b_{k+1})+3(a_{k}+b_k)\\
&=(a_{k+2}-5a_{k+1}+3a_{k})+(b_{k+2}-5b_{k+1}+3b_{k})\\
&=0+0=0.
\end{align*}
Thus the sum $(a_i)_{i=1}^{\infty}+(b_i)_{i=1}^{\infty}=(a_i+b_i)_{i=1}^{\infty}$ satisfies the recurrence relation and it is also in $U$. Hence condition 2 is met.
To check condition 3, take $(a_i)_{i=1}^{\infty} \in U$ and let $c\in \R$ be a scalar.
Then the sequence $(a_i)_{i=1}^{\infty}$ satisfies (*). Multiplying (*) by $c$, we have
\[(ca_{k+2})-5(ca_{k+1})+3(ca_{k})=0.\]
This implies that the scalar product $c(a_i)_{i=1}^{\infty}=(ca_i)_{i=1}^{\infty}$ satisfies the recurrence relation, and hence it is in $U$. Thus condition 3 is satisfied.
We have checked all subspace criteria, and thus $U$ is a subspace of the vector space $V$.
Related Question.
This is the first problem of three problems about a linear recurrence relation and linear algebra.
- [Problem 2] The problems/solutions of finding a basis and dimension of $U$ and finding a matrix representation of some linear transformation of $U$ to itself are given in the post
Matrix representation of a linear transformation of subspace of sequences satisfying recurrence relation. - [Problem 3] Want to know how to find a general formula for a sequence satisfying a linear recurrence relation using linear algebra? Check out the post Solve linear recurrence relation using linear algebra (eigenvalues and eigenvectors)
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