I'm asked to show if the following sets are a subspace of $S$, the space of sequences.
$$B_1=\{(a_n)_n \in S|a_n=2a_{n-1}-3a_{n-2} \text{ for all } n \geq 2 \}$$
$$B_2=\{(a_n)_n \in S|a_n = 0 \text{ whenever 3 divides $n$} \}$$.
I know the properties of a subspace.
$1$. The zero vector has to be defined.
$2$. Addition has to be closed for this set.
$3$. Scalar multiplication has to be closed for this set.
I can verify addition very easily.
How would I show that if the $0$ vector and scalar multiplication is defined though? I'm not told the first term of the sequence so I'm not really sure how the various properties can be verified.
Can someone provide a hint of some kind? Thanks!