condition for upper triangular matrix

Question

Consider the following condition from this other post

1. Define $S_k = \operatorname{span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular if and only if $T S_k \subset S_k$.

From this condition, wouldnt any linear map of dimension $\leq k$ be upper triangular? If not, what am I not understanding?

Answer 1

The correct criterion is the following:

An $n\times n$ matrix $T$ is upper triangular if and only if $TS_k \subseteq S_k$ for all $k\in\{1,\ldots,n\}$.