Verify the subspace of an infinite vector space is finite-dimensional

linear algebra

so on a homework for my linear algebra course, I am stuck with the following.
There is a subspace
$S=\{(x_1,x_2,…): x_n+x_{n+1}-x_{n+2}=0, \forall n \geq 1\} \subseteq \mathbb{F}^\infty$
My task is to answer if (a) S is finite dimensional, and then (b) what its dimension is.
My guess so far is to show that if $S$ is finite dimensional, then the list of vectors $(x_1,x_2,…)$ spans the set, which will generally contain vectors of the form $(v_1,v_2,…)$, for $\mathbb{F}^\infty$. Thus I somehow need to show that
$\bigoplus_{i=1}^{\infty} c_ix_i=\sum_{j=1}^{\infty} v_j$
but this creates an infinite system of equations I don't know how to solve. Tips? Thank you

Best Answer

Let $a$ and $b$ be any two numbers and consider the sequence $(x_n)_{n\in\Bbb N}$ such that:

  • $x_1=a$;
  • $x_2=b$;
  • $(\forall n\in\Bbb N):x_{n+2}=x_n+x_{n+1}$.

Then $(x_n)_{n\in\Bbb N}\in S$ and every element of $S$ is of this type (with $a=x_1$ and $b=x_2$). So, the whole space $S$ depends upon the two parameters $a$ and $B$, which suggests that $\dim S=2$. Can you take it from here?

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