Is it true that the conditions that define any proper subspace $S$ of a vector space $V$, with $\dim(V)<+\infty$, can be express as a homogeneous system of linear equation?
Let $S=\{(x,y)\in\mathbb{R}^2: e^{x}=1\}$, the condition that defines $S$ is the non linear equation $e^{x}=1$, but it is equivalent to $x=0$, so $S=\{(x,y)\in\mathbb{R}^2: x=0\}$ ($S$ is clearly a subspace of $\mathbb{R}^2$). Another expample: let $$S=\{(x,y)\in\mathbb{R}^2: x^2-2x y+y^2=0\}$$
In this case, the condition that defines the elements of $S$ is $x^2-2x y+y^2=0$ witch is equivalent to the equation $(x-y)^2=0\ \ \ \to \ \ \ x-y=0$, so $S$ can be express as $$S=\{(x,y)\in\mathbb{R}^2: x-y=0\}$$ and it's easy to prove that it is a subspace of $\mathbb{R}^2$.
It seems to me that the response is positive if $V=\mathbb{R}^n$, but what can I say if $V$ is a polynomial vector space, or a matrix vector space? I know, maybe it's a silly question, but I would like to know that it's true or not. Thanks.