- #1

- 775

- 1

## Main Question or Discussion Point

I've been thinking about a problem I made up. The solution may be trivial or very difficult as I have not given too much thought to it, but I can't think of an answer of the top of my head.

Let ## T:V → V ## be a linear operator on a finite-dimensional vector space ##V##. Does there exist a vector ## v \in V ## for which the T-cyclic subspace of ##V## generated by ##v## is ##V##? This is certainly not true in general, since if ##T## is the zero transformation and ##V## has dimension greater than 1 then no T-cyclic subspace will equal ##V##.

But what about for an arbitrary linear map?

BiP

Let ## T:V → V ## be a linear operator on a finite-dimensional vector space ##V##. Does there exist a vector ## v \in V ## for which the T-cyclic subspace of ##V## generated by ##v## is ##V##? This is certainly not true in general, since if ##T## is the zero transformation and ##V## has dimension greater than 1 then no T-cyclic subspace will equal ##V##.

But what about for an arbitrary linear map?

BiP