Infinitely many subspaces in R3 ?

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In summary, the conversation discusses the concept of subspaces in R3 and how they can be expressed using specific examples. It is stated that there are infinitely many lines and planes containing the origin, but R3 only has one 3-dimensional subspace. The conversation also addresses the misconception that a vector space always has 0 as the zero vector. It is clarified that this is only true for the usual definitions of scalar multiplication and vector addition.
  • #1
Mimi
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infinitely many "subspaces" in R3 ?

In R3, there are zero, 1, 2, 3 dimensional subspaces. But how can I express them with 'specific' example, using variables x,y,and z?
 
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  • #2
Mimi said:
In R3, there are zero, 1, 2, 3 dimensional subspaces. But how can I express them with 'specific' example, using variables x,y,and z?

There are infinite many lines through the origin.

That's a hint.
 
  • #3
And there are infinitely many planes containing the origin! But R3 has only one 3 dimensional subspace.

If by "specific example" you mean a specific example of each: what is the equation of a line through the origin? What is the equation of a plane containing the origin? And as I said, you don't have many choices for an example of a 3 dimensional supspace.
 
  • #4
In addition, know that every subspace of R^n contains the origin.
 
  • #5
benorin said:
In addition, know that every subspace of R^n contains the origin.

That's false.

You are assuming the zero vector to be the origin. It doesn't have to be.
 
  • #6
I did mean to state this only for the usual definitions of scalar multiplication and vector addition; in such a case, R^3 has precisely these subspaces: the trivial ones, namely, the set containing the zero vector and R^3 itself, (which are the zero and 3 dimensional subspaces); each line containing the origin is a subspace of dimension 1 (and there are no others of dimension 1); and lastly, each plane containing the origin is a subspace of dimension 2 (and there are no others of dimension 2).
 
  • #7
benorin said:
I did mean to state this only for the usual definitions of scalar multiplication and vector addition; in such a case, R^3 has precisely these subspaces: the trivial ones, namely, the set containing the zero vector and R^3 itself, (which are the zero and 3 dimensional subspaces); each line containing the origin is a subspace of dimension 1 (and there are no others of dimension 1); and lastly, each plane containing the origin is a subspace of dimension 2 (and there are no others of dimension 2).

That's better. :biggrin:

I got a True/False question wrong once because I answered it too quickly and it was regarding this. Not directly, but in general.

So, for the sake of being concise, I always try to add the conditions when I make a comment or statement. I forget sometimes, but that's normal.

The reason why I pointed it out here is because the thread starter is relatively new to linear algebra, and he/she should get a feel of what a vector space is. People get mistaken that a vector space actually has 0 as the zero vector all the time.
 
  • #8
The original post said "subspace of Rn". That is sufficient to conclude that the 0 vector is (0,0). I'm not sure what you mean by "People get mistaken that a vector space has 0 as the zero vector all the time". Are you thinking about a case in which there is an unusual definition of vector addition? Precisely because "the thread starter is relatively new to linear algebra" I think bringing up situations like that when it was clearlys stated that this is in Rn will be confusing.
 

1. What is a subspace in R3?

A subspace in R3 is a subset of R3 that satisfies three properties: it contains the zero vector, it is closed under addition, and it is closed under scalar multiplication. In other words, it is a space within R3 that is closed under certain operations.

2. How many subspaces can exist in R3?

There are infinitely many subspaces that can exist in R3. This is because for any given subspace in R3, we can always find another subspace that is a subset of it, making the number of possible subspaces infinite.

3. What is the dimension of a subspace in R3?

The dimension of a subspace in R3 is the number of linearly independent vectors that span the subspace. In other words, it is the number of vectors needed to form a basis for the subspace.

4. Can a subspace in R3 contain more than three dimensions?

No, a subspace in R3 can only contain a maximum of three dimensions. This is because R3 is a three-dimensional space, so any subspace within it can only have a maximum of three dimensions.

5. How are subspaces in R3 related to linear transformations?

Subspaces in R3 are closely related to linear transformations. In fact, every linear transformation in R3 can be represented as a matrix, and the image of the linear transformation is a subspace in R3. Additionally, the null space of the matrix representation of the linear transformation is also a subspace in R3.

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