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

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In summary, the conversation discussed the possible subspaces of vector spaces R2 and R3, including the subspaces of {0}, R2, all lines through the origin, and all planes through the origin. There was a question about proving the fourth subspace in R3, but no clear answer was provided. The suggestion was made to find the equation of a plane through the origin and show that the set of all points that satisfy this equation is a subspace of R3.
  • #1
pr0me7heu2
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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?
 
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  • #2
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
 

1. What is a subspace in R2 and R3?

A subspace in R2 and R3 is a subset of R2 or R3 that contains all linear combinations of its vectors. This means that if you take any two vectors in the subspace and add them together, the result will also be in the subspace. Additionally, multiplying any vector in the subspace by a scalar will also result in a vector that is still in the subspace.

2. How do you determine if a set of vectors is a subspace in R2 or R3?

To determine if a set of vectors is a subspace in R2 or R3, you can use the three properties of a subspace: closure under addition, closure under scalar multiplication, and the inclusion of the zero vector. If the set of vectors satisfies all three properties, then it is a subspace.

3. Can a subspace in R2 or R3 have more than two dimensions?

No, a subspace in R2 or R3 can only have two dimensions. This is because R2 and R3 are two and three-dimensional vector spaces, respectively. Any subspace within these spaces can only have dimensions that are equal to or less than the dimensions of the original space.

4. How do subspaces in R2 and R3 relate to linear independence?

Subspaces in R2 and R3 are made up of linearly independent vectors. This means that none of the vectors in the subspace can be written as a linear combination of the others. Additionally, a set of vectors is considered a basis for a subspace if it is both linearly independent and spans the subspace.

5. Can subspaces in R2 and R3 intersect?

Yes, subspaces in R2 and R3 can intersect. In fact, the intersection of two subspaces is itself a subspace. However, the intersection can also be the empty set, which is still considered a subspace. The intersection of two subspaces is typically denoted as S ∩ T.

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