I need to prove that $U = \{x\in\mathbb{R^2} : x_1^2 + x_2 = 1\}$ is not an affine linear subspace of $\mathbb{R^2}$.
Every vector $(x_1,x_2)$ from $U$ satisfies the relation: $x_1^2 + x_2 = 1$.
Let's suppose that $U$ is an affine linear subspace of $\mathbb{R^2}$. That would mean that there exist:
- Subspace of $\mathbb{R^2}$ , let's call it $V$.
- Other vector $x$ from $\mathbb{R^2}$
Such that $V + x = U$ ( Every vector from $V$ added with $x$ gives us $U$ ). To prove that this is not an affine linear subspace, I would need to try and find the needed $V$ and $x$ and then come to a conclusion that $x$ must also belong to $V$. That would mean that the rules are not satisfied and that $U$ is not an affine linear subspace. How do I pick the right $V$ and $x$ ?
I have an equation: $V + x = U$ . Here I have $2$ unknown vectors. How do I choose the right subspace and vector so that I can prove what I want to prove ??
Hint:
Step 1: how are the subspace of $R^2 $?
Answer : $\{0\}$, $L:=\lambda v$ and $R^2 $
Step 2: prove that your U can't be the previous subspaces plus any x