Null space of infinite-dimensional matrix

Question

Is there a general way to tell if the null space of an infinite-dimensional matrix contains a vector that is not zero?

This question connects to the following problem:

If I know that

\begin{equation} \sum_{n=0}^\infty c_{a,b} (n) f(n) =0 \end{equation}

for $a,b$ any positive integer, where

\begin{equation} c_{a,b}(n) = 2^a \binom{n}{a} - 2^b \binom{n}{b}\ , \end{equation}

what is $f(n)$?

I know for certain that this $c_{a,b}(n)$ has a non trivial solution. Is there a way to determine if a different $c_{a,b}(n)$ also has a non trivial solution?

If so, is there a general strategy I can use to solve these types of problems (even for different sequences $c_{a,b}(n)$)?

Answer 1

This is a long comment. You have a question, a problem, and two more questions.

1. "Is there a general..." Certainly not, if general means always works.

2. But the problem is about linear dependence of a sequence of matrices $C_n$ that have $c_{ab}(n)$ in the upper left $n$ by $n$ block and zeros out to infinity. Since $C_1=C_2=0$ the equation $$ \sum f(n)C_n=0 $$ is satisfied with $f(1)$ and $f(2)$ arbitrary and the rest 0.

3. "...another $c_{ab}$ ..." needs to be more specific.

4. "...these types of problems..." Maybe Yes, if all the matrices $C_n$ are finite blocks followed by zeros as above, you can try dealing with a finite number of them as in item 2.