Is the set of matrices $X \in M_{n\times n}$ satisfying $AX + XA^T = -L$ for a given but arbitrary $A,L \in M_{n\times n}$ a subspace? Is the set of matrices $X \in M_{n\times n}$ satisfying $AX + XA^T = -L$ for a given but arbitrary $A,L \in M_{n\times n}$ a subspace under usual definitions of matrix addition and scalar multiplication?
Apparently it is not ... For $X_1, X_2\in M_{n\times n}$ and $c \in R$ it is clearly closed by addition and multiplication due to matrix operations. The zero vector is not obvious to me since if $X=[0]$, this would have to make $L=0$ which may not be the case. Is that enough of a justification?

You are right. It is a subspace only if $L=0$. Suppose $X_1$ and $X_2$ belong to the set. Then $AX_1+X_1A^T=-L$ and $AX_2+X_2A^T=-L$ and it follows that $A(X_1+X_2)+(X_1+X_2)A^T=-2L$, but $-2L=-L$ only if $L=0$. In general, non-homogeneous linear relations do not lead to (vector) subspaces.
