Subset of the domain for the transformation to be invariant

[songoku]

Homework Statement

Given transformation matrix M : $\mathbb R^2 \rightarrow \mathbb R^2$
$$\begin{pmatrix}
-1 & 0\\
0 & 1
\end{pmatrix}
$$

Describe the subset of the domain for the transformation to be invariant

Relevant Equations

$M \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} x\\ y \end{pmatrix}$

I found that the
a) invariant points are all points on y-axis
b) invariant lines are y-axis and $y=c$ where $c$ is real

I am confused what the final answer should be. How to state the answer as "subset of domain"? Is it:
$$\{x,y \in \mathbb R^2 | (0, y) , x = 0, y=c\}$$

Thanks

 

[PeroK]

The invariant points are the y-axis. You can express this in set notation a number of ways. Perhaps the simplest is:
$$\{(x,y) \in \mathbb R^2 | x = 0\}$$PS That's for part a).

 

[fresh_42]

Homework Statement:: Given transformation matrix M : $\mathbb R^2 \rightarrow \mathbb R^2$
$$\begin{pmatrix}
-1 & 0\\
0 & 1
\end{pmatrix}
$$

Describe the subset of the domain for the transformation to be invariant
Relevant Equations:: $M \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} x\\ y \end{pmatrix}$

I found that the
a) invariant points are all points on y-axis
b) invariant lines are y-axis and $y=c$ where $c$ is real

I am confused what the final answer should be. How to state the answer as "subset of domain"? Is it:
$$\{x,y \in \mathbb R^2 | (0, y) , x = 0, y=c\}$$

Thanks

You are right that there are two different types of sets, the y-axis, and all horizontal lines because the transformation is a reflection at the y-axis. So the question is problematic as it asks for one set when there are infinitely many. There are even four types because the entire space is invariant, too. And the word subspace normally does not exclude equality. And then there is the zero.

My answer would be
$$
\{(0,0)\}\, , \,\{(x,y)\in \mathbb{R}^2\,|\,x=0\} \, , \,\{(x,y)\in \mathbb{R}^2\,|\,y=c\}\;(c\in \mathbb{R})\, , \,\mathbb{R}^2.
$$

 

[fresh_42]

The invariant points are the y-axis. You can express this in set notation a number of ways. Perhaps the simplest is:
$$\{(x,y) \in \mathbb R^2 | x = 0\}$$

Invariant points (eigen vectors) and invariant subspaces are two different things.

 

[PeroK]

You are right that there are two different types of sets, the y-axis, and all horizontal lines because the transformation is a reflection at the y-axis. So the question is problematic as it asks for one set when there are infinitely many. There are even four types because the entire space is invariant, too. And the word subspace normally does not exclude equality. And then there is the zero.

My answer would be
$$
\{(0,0)\}\, , \,\{(x,y)\in \mathbb{R}^2\,|\,x=0\} \, , \,\{(x,y)\in \mathbb{R}^2\,|\,y=c\}\;(c\in \mathbb{R})\, , \,\mathbb{R}^2.
$$

This I don't understand. For part b), unless we use a formalism for sets of lines, I wouldn't use set notation. I'd just say "the set of lines in the plane defined by $y = c$, where $c \in \mathbb R$; and the y-axis".

 

[fresh_42]

This I don't understand. For part b), unless we use a formalism for sets of lines, I wouldn't use set notation. I'd just say "the set of lines in the plane defined by $y = c$, where $c \in \mathbb R$; and the y-axis".

Yes, my fault. Only $\{(x,0)\}$ is an invariant subspace as $\{(x,c)\}$ are no subspaces, only invariant sets, But invariant subsets are myriads more, every figure that coincides with its mirror image. The horizontal lines are only affine subspaces.

 

[songoku]

Thank you very much PeroK and fresh_42