Subspace of R^3

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December 28th 2010, 03:59 AM
chris2547

Subspace of R^3

Hello,

I was wondering if $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$ ? Some classmates of me tell me it is, some say it isn't.

I'm under the impression that it is in fact a subspace, anyone care to elaborate?

Thanks in advance
December 28th 2010, 04:19 AM
Ackbeet

Sure it is. $\mathbb{R}^{2}$ is a vector space in its own right, and it is a subset of $\mathbb{R}^{3}.$ Ergo, it's a subspace. The only confusion might be in a mix of notations. But you can just say, for example, that the third component is always zero (sort of an embedding scheme, I guess).

Does that help?
December 28th 2010, 04:58 AM
FernandoRevilla

Quote:

Originally Posted by chris2547

I was wondering if $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$ ? Some classmates of me tell me it is, some say it isn't.

A necessary condition to be $F$ subspace of $E$ is that $F\subset E$ . Taking into account that $\mathbb{R}^2\not\subset \mathbb{R}^3$ we can't say strictly that $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$ .

However (and following Ackbeet's outline) you can define:

$f:\mathbb{R}^2\rightarrow \mathbb{R}^3\;\quad f(x_1,x_2)=(x_1,x_2,0)$

This function is an monomorphism (with usual operations) , so we can identify algebraically $\mathbb{R}^2$ with $\textrm{Im}f\subset \mathbb{R}^3$ (which is a subspace of $\mathbb{R}^3$ . In that sense we say " $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$ " .

Fernando Revilla
December 28th 2010, 04:58 AM
DrSteve

Just to clarify what Ackbeet is saying:

Technically it is not a subspace, but it is isomorphic to one in a very strong sense (by adding an extra component which is always zero).

Most mathematicians will say it is a subspace by thinking in terms of this isomorphism (they are technicaly wrong by saying this, but this convention has been agreed upon in the mathematical community).