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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:

  1. Subspace of $\mathbb{R^2}$ , let's call it $V$.
  2. 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 ??

1 Answers 1

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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

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    I'm having a bit of trouble with the proof for the L subspace. Specifically, x and u shouldn't belong to the same line. That means that if I represent the vectors as v:= (v1,v2) and u:=(u1,u2) , I have v2 = m*v1 and u2 = n*u1 with m not equal to n. Now if I add them and multiply them by some lambdas and apply the given relation for the affine subtree I get a huge equation. Could you give me a hint ? – 2017-01-09
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    @db9117 try to prove this: for each x , v, then U\cap L_v + x has 0 , 1 or 2 elements – 2017-01-09
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    Could you be more specific, please? I've tried everything I could think of, so far no luck – 2017-01-10