This might be easier than I think, but I got stuck.
Assume a vector $y=[y_1,\ldots,y_n]\in Y$, where $Y$ is a convex polyhedron. Assume a $k$-dimensional subvector of $y$, namely $y^K=[y_1,\ldots,y_k]$.
I need to say something about the domain of $y^K$. Say that $y^K\in Y^K$, what can I say about the relationship between $Y^k$ and $Y$? Is $Y^K$ a subspace of $Y$?
Strictly speaking $Y_k$ ( for $k<n$) is not a subspace of Y.
A vector space is defined as a quadruple $Y=(V,K,+,\cdot)$ where $V$ is a set, $K$ a field, and $+:V\times V \rightarrow V$ and $\cdot: K\times V \rightarrow V$ are operations that satisfies certain axioms (see here).
A subspace is defined as a subset of $V$ that is a vector space with the same field $K$ and the same operations.
In your case the set $V$ for the space $Y$ is the set of $n-$ples $(y_1,\cdots, y_n)^T$ and the elements of $Y_k$ are $k-$ples , so $Y_k$ is not a subset of $V$. But the subsespaces of $Y$ of dimension $K$ are isomorphic to $Y_k$ (if the field $K$ and the operations are the same).