# Why R2 is not a subspace of R3?

1. Aug 25, 2006

### Bob

I think R2 is a subspace of R3 in the form(a,b,0)'.

2. Aug 25, 2006

### shmoe

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. Aug 25, 2006

### HallsofIvy

Staff Emeritus
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. Oct 18, 2009

### MaxManus

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. Oct 18, 2009

### lanedance

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
$$(a,b) \in \mathbb{R}^2 \leftrightarrow (a,b, 0) \in \mathbb{R}^3$$

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

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
$$(a,b) \in \mathbb{R}^2 \leftrightarrow (a,b,1) \in \mathbb{R}^3$$
then clearly the zero vector is not in the embedded R^2, so it is not a subspace of R^3

Last edited: Oct 18, 2009
6. Oct 18, 2009

### HallsofIvy

Staff Emeritus
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!

Last edited: Oct 18, 2009
7. Oct 18, 2009

### MaxManus

Thanks to both of you.