True/False: $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$?

Question

The answer is False.

Since $\mathbb{R}^2$ is not a subset of $\mathbb{R}^3$.

However, can someone explicitly define the difference between a subset and subspace? I am somewhat struggling with the technicality of the following definition

Subspace: a subspace, U of a vector space V, is a subset of V, $U \subset V$, such that U is a vector space in its own right, over the same field.

Vector space: A vector space over a field is a set $V$ of objects that is closed under binary operation ('addition') and scalar multiplication

Also, what would make $\mathbb{R}^2$ a subspace of $\mathbb{R}^3$ if anything?

Thanks!

Answer 1

The set of things in $\Bbb{R}^3$ is $3$-component real vectors. The set of thing in $\Bbb{R}^2$ is $2$-component real vectors. (The component count is exactly what is being communicated by the "$n$" in $\Bbb{R}^n$.) Since there are no $2$-component vectors in $\Bbb{R}^3$, no element of $\Bbb{R}^2$ is in $\Bbb{R}^3$. So $\Bbb{R}^2$ is not a subset, and hence not a subspace, of $\Bbb{R}^3$.

Answer 2

Any set of elements $\in \mathbb R^3$ is by definition a subset of $\mathbb R^3$. A subspace is a particular subset which is a vector space, that is it is closed by multplication by scalars and by addition.

Therefore since we have that for any

$$\vec v\in \mathbb R^2 \implies \vec v \not \in \mathbb R^3 $$

then $\mathbb R^2$ can't be neither a subset nor a subspace of $\mathbb R^3 $.

What is true is that any plane through the origin is a subspace of $\mathbb R^3$.

Answer 3

When we say that $\mathbb R^2$ is a subspace of $\mathbb R^3$, we mean the canonical embedding of $\mathbb R^2$ into $\mathbb R^3$, i.e., $(a,b)\mapsto(a,b,0)$ is a subspace.

It's not hard to show that this is true.

Answer 4

$\mathbb{R}^2$ is neither a subset nor a subspace of $\mathbb{R}^3$, but $\mathbb{R}^2$ is isomorphic to a subspace of $\mathbb{R}^3$. Every subspace is also a subset and by definition, $\mathbb{R}^2$ cannot be a subspace of $\mathbb{R}^3$. In general, let $V$ be a vector space and $U$ be a subset of $V$. We say that $U$ is a subspace of $V$ if $U$ is also a vector space (so, in short, remember that a subspace is a subset of a larger vector space that is a vector space as well).

Answer 5

To answer the question you ask, a "subset" of a vector space is a "subspace" if and only if it is "closed" under vector addition and scalar multiplication. That is, if u and v are in the subset so is u+ v and av for any number a.

However, that is not really what you want to know! R^2 is the set of vectors of the form (a, b) while R^3 nis the set of vectors of the form (a, b, c). The first is not even a subset of the latter! (There is an isomorphism from (a. b) to (a, b, 0) which IS a subspace of R^3.)