- Thread starter
- #1

- Apr 14, 2013

- 4,036

We have the following subsets:

\begin{align*}&U_1:=\left \{\begin{pmatrix}x \\ y\end{pmatrix} \mid x^2+y^2\leq 4\right \} \subseteq \mathbb{R}^2\\ &U_2:=\left \{\begin{pmatrix}2a \\ -a\end{pmatrix} \mid a\in \mathbb{R}\right \} \subseteq \mathbb{R}^2 \\ &U_3:=\left \{\begin{pmatrix}x \\ y \\ z\end{pmatrix} \mid y=0\right \}\subseteq \mathbb{R}^3 \\ &U_4:=\left \{\begin{pmatrix}x \\ y \\ z\end{pmatrix} \mid y=1\right \}\subseteq \mathbb{R}^3\end{align*}

I want to sketch these sets and check in that way if these are subspaces.

We have the following graphs:

- $U_1$ :

This is a subspace, isn't it? But how can we explain that from the graph?

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- $U_2$ :

Since this line goes through the origin, it is a subspace, or not?

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- This is the $xz$ plane with $y=0$.

This is a subspace, since the zero vector is contained, or not?

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- This is the $xz$ plane with $y=1$.

This is not a subspace since the zero vector is not contained. Is that correct?