Let \(X\) be a separable infinite-dimensional Banach space. A recent new notion in linear dynamics was introduced by \textit{B. F. Madore} and \textit{R. A. Martínez-Avendaño} in [J. Math. Anal. Appl. 373, No. 2, 502--511 (2011; Zbl 1210.47023)], namely, the notion of subspace-hypercyclicity. A bounded linear operator \(T:X\rightarrow X\) is called \textit{subspace-hypercyclic for a subspace \(M\)} of X if there exists a vector \(x\in X\) such that \(\{T^n x:n\in\mathbb{N}\}\cap M\) is dense in \(M\). When \(M=X\), this is just the definition of hypercyclicity, and the operator \(T\) is said to be hypercyclic. \textit{P. S. Bourdon} [Proc. Am. Math. Soc. 118, No. 3, 845--847 (1993; Zbl 0809.47005)] and \textit{D. A. Herrero} [J. Funct. Anal. 99, No. 1, 179--190 (1991; Zbl 0758.47016)] proved that, given any hypercyclic operator \(T\) on \(X\), there is a dense invariant subspace consisting, except for zero, of hypercyclic vectors. The real case was analyzed by \textit{J. P. Bès} [Proc. Am. Math. Soc. 127, No. 6, 1801--1804 (1999; Zbl 0914.47005)]. These results depend on the fact that, if \(T\) is a hypercyclic operator, then \(p(T)\) has dense range for any polynomial \(p\in\mathbb{C}[z]\setminus\{0\}\). The existence of a similar result for subspace-hypercyclicity was asked by B. F. Madore and R. A. Martínez-Avendaño in the aforementioned paper. In this note, a positive answer is provided by the author. In addition, this allows to describe the algebraic structure of the set of subspace-hypercyclic vectors for a subspace-hypercyclic operator \(T\) on \(M\).