Subspaces of R2 and R3: Understanding Dimensions of Real Vector Spaces

[pr0me7heu2]

So I'm considering dimensions of real vector spaces.

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?

 

[ircdan]

what is the equation of a plane through the origin? you should show that the set consisting of all points that lie on this plane(ie, satisfy this equation once you get it) is a subspace of R^3