We have the subsets of $\mathbb{R}^2$: $X_1 := \{(x,y) \in \mathbb{R}^2 : x + y = 1\}, X2 := \{(x,y) \in \mathbb{R}^2 : x^2- y^2 = 0\}$. I have shown that these are not a linear subspace of the $\mathbb{R}$-vector space $\mathbb{R}^2$.
The span $\langle X_i\rangle$ contains every linear combination of the elements of $X_i$, right?
We have that $X_1 := \{(x,y) \in \mathbb{R}^2 : x + y = 1\}=\{(x,y) \in \mathbb{R}^2 : y = 1-x\}=\{(x,1-x) \mid x\in \mathbb{R}\}=\{(0,1)+x(1,-1)\mid x\in \mathbb{R}\}$