- #1

mathmari

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MHB

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The three axioms for a subspace are:

S1. The set must be not-empty.

S2. The sum of two elements of the set must be contained in the set.

S3. The scalar product of each element of the set must be again in the set.

I have shown that:

- $\displaystyle{X_1=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid 5x+3y=0_{\mathbb{R}}\right \}}$ is a subspace.

All axioms are satisfied.

- $\displaystyle{X_2=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid 5x+3y=-2\right \}}$ is not a subspace.

S1 is satisfied, S2 and S3 are not satisfied.

- $\displaystyle{X_3=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid x^2+y^2=0_{\mathbb{R}}\right \}=\left \{\begin{pmatrix}0 \\ 0 \end{pmatrix}\right \}}$ is a subspace.

All axioms are satisfied.

- $\displaystyle{X_4=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid x^2+y^2=5\right \}}$ is not a subspace.

Only S1 is satisfied.

- $\displaystyle{X_5=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid x,y\in \mathbb{Z}\right \}}$ is not a subspace.

S3 is not satisfied, S1 and S2 are satisfied.

- $\displaystyle{X_6=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid xy\geq 0_{\mathbb{R}}\right \}}$ is not a subspace.

S3 is not satisfied, S1 and S2 are satisfied.

- $\displaystyle{X_7=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb{R}^2\mid x> 0_{\mathbb{R}}<y\right \}}$ is not a subspace.

S3 is not satisfied, S1 and S2 are satisfied.

Give all subsets of $\mathbb{R}^2$ that

- satisfy S2 and S3, but S1

- satisfy S3, but S1 and S2

Since they shouldn't satisfy S1 do we not have only the empty set? :unsure: