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I found myself thinking about the following:

So for the vector space R2 there are the following possible subspaces:

1. {0}

2. R2

3. All the lines through the origin.

Then I considered R3.

For the vector space R3 there are the following subspaces:

1. {0}

2. R3

3. All lines through the origin.

4. All planes through the origin.

Although I "know" (4.) to be true... I can't figure out a mathematical why or a solid way of proving it.

Any hints?