Why R2 is not a subspace of R3?

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In summary: I think the key here is, before you can discuss whether elements of R^2 are closed under addition in R^3, you first need to know how you map an element of R^2 into R^3 (the isomorphism). If you've done that, you should be able to show using the three subspace criteria, that R^2 is isomorphic to a subspace of R^3.
  • #1
Bob
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I think R2 is a subspace of R3 in the form(a,b,0)'.
 
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  • #2
R^2 is isomorphic to the subset (a,b,0) of R^3, but it's also isomorphic to infinitely many other subspaces of R^3 (any 2 dimensional one). As such, there's no canonical embedding, and you don't usually think of R^2 as being contained in R^3.

A more obvious explanation is the vector (a,b) is not the same as the vector (a,b,0). 2 components vs 3 components, so they are different objects.
 
  • #3
Shmoe is correct. However, it is common to speak of isomorphic things as if they were the same thing. Most mathematicians would say (with "abuse of terminology") that R2 is a subspace of R3, understanding that what they really mean is that it is isomorphic to one.
 
  • #4
I know that it is an old thread, but I still don't get why R^2 is not a subspace of R^3. Is it only because R^3 has 3 components and R^2 only 2 components? Is it possible to use the three conditions to show that R^2 is not a subspace of R^3?
1. The zero vector, 0, is in W.
2. If u and v are elements of W, then the sum u + v is an element of W;
3. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;
 
  • #5
I think the point in the threads above is that R^2 & R^3 are different objects, before you can discuss whether R^2 is a subspace of in R^3 you need to "embed" R^2 in R^3 by defining an isomorphism between a subset of R^3 & all of R^2, the obvious one being
[tex] (a,b) \in \mathbb{R}^2 \leftrightarrow (a,b, 0) \in \mathbb{R}^3 [/tex]

however as schmoe pointed out there are infinite ways to do it eg. another isomorphsim toa subpsapce of R^3 is
[tex] (a,b) \in \mathbb{R}^2 \leftrightarrow (a,0,b) \in \mathbb{R}^3 [/tex]

i think the key here is, before you can discuss whether elements of R^2 are closed under addition in R^3, you first need to know how you map an element of R^2 into R^3 (the isomorphism)

if you've done that, you should be able to show using the 3 subspace criteria, that R^2 is isomorphic to a subspace of R^3. Then as pointed out, many people would be happy to accept the abuse of terminology and say R^2 is a subspace of R^3, implying there is an isomorphism to a subspace of R^3

however, as an example what if you chose to embed R^2 in R^3 by
[tex] (a,b) \in \mathbb{R}^2 \leftrightarrow (a,b,1) \in \mathbb{R}^3 [/tex]
then clearly the zero vector is not in the embedded R^2, so it is not a subspace of R^3
 
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  • #6
MaxManus said:
I know that it is an old thread, but I still don't get why R^2 is not a subspace of R^3. Is it only because R^3 has 3 components and R^2 only 2 components? Is it possible to use the three conditions to show that R^2 is not a subspace of R^3?
1. The zero vector, 0, is in W.
2. If u and v are elements of W, then the sum u + v is an element of W;
3. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;
To begin with, for W to be a subspace of V, it must be a subset of V. Things in R^2 are of the form (a, b), with two components while things in R^3 are of the form (a, b, c) with three components. Members of R^2 are not members of R^3 so R^2 is not a subset of R^3.

That said, originally, I was a little surprised by the question. It is common to think of R^2 as being a subset of R^3 using the obvious isomorphism to a subspace of R^3: (a, b)-> (a, b, 0). Strictly speaking, it is not R^2 that is a subspace of R^3, it is that subspace. But one has to very strict!
 
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  • #7
Thanks to both of you.
 

1. Why is R2 not a subspace of R3?

R2 is not a subspace of R3 because the two spaces have different dimensions. R2 is a two-dimensional vector space, while R3 is a three-dimensional vector space. This means that the two spaces have different bases and therefore, vectors in R2 cannot be represented in R3 and vice versa.

2. Can a subset of R3 be a subspace of R2?

No, a subset of R3 cannot be a subspace of R2. A subspace must satisfy the three conditions of closure under addition, closure under scalar multiplication, and contain the zero vector. Since R2 and R3 have different dimensions, a subset of R3 cannot satisfy these conditions and therefore, cannot be a subspace of R2.

3. What is the difference between R2 and R3?

The main difference between R2 and R3 is their dimensions. R2 is a two-dimensional vector space, while R3 is a three-dimensional vector space. This means that R3 has one more dimension than R2, and therefore, can represent more complex and diverse mathematical concepts.

4. Can a subspace of R3 be a subspace of R2?

No, a subspace of R3 cannot be a subspace of R2. This is because a subspace must satisfy the three conditions of closure under addition, closure under scalar multiplication, and contain the zero vector. Since R2 and R3 have different dimensions, a subspace of R3 cannot satisfy these conditions and therefore, cannot be a subspace of R2.

5. How does the number of dimensions affect vector spaces?

The number of dimensions affects vector spaces by determining the maximum number of independent vectors that can exist in that space. For example, R2 can have a maximum of two independent vectors, while R3 can have a maximum of three independent vectors. This also affects the types of operations and transformations that can be performed in each space.

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