Let $k<d\in\mathbb{N}$. Given the following definition:
$G=\{ S: S\text{ is }k\text{-dimensinal subspace of }\mathbb{R}^d\}$
Would you understand that $G$ contains only "homogeneous linear subspaces" of dimension $k$, or all linear subspaces of dimension $k$?
For example if $d=2, k=1$ would $G$ contain the 1-dimensinal subapace $\begin{pmatrix}1\\3\end{pmatrix}+t\begin{pmatrix}1\\7\end{pmatrix}$?
A linear subspace must itself be a vector space. However, vector spaces must contain the additive identity which in the case of $\mathbb{R}^n$ is just the $0$ vector. So any subspaces of $\mathbb{R}^n$ must contain the $0$ vector, which means that the $\textit{only}$ linear subspaces of $\mathbb{R}^n$ are "homogeneous linear subspaces."