$R^2$ is subspace of $R^3$ or not some author consider $R^2$ is subspace of $R^3$ but some author say $R^2$ is not a subspace of $R^3$.
so when they consider in which sense they consider?
 A: The vector space $\mathbb R^2$ can be embedded into $\mathbb R^3$, that is, it is isomorphic to a subspace of $\mathbb R^3$. Actually there are infinitely many ways to embed $\mathbb R^2$ into $\mathbb R^3$, but they are actually equivalent (that is, for any two embedddings of $\mathbb R^2$ to $\mathbb R^3$ there's an automorphism of $\mathbb R^3$ mapping those embeddings to each other).
Moreover there's a particularly simple embedding, which just maps the $k$-th basis vector of the standard basis of $\mathbb R^2$ to the $k$-th basis vector of the standard basis of $\mathbb R^3$, resulting in the map
$$\pmatrix{x\\y}\mapsto\pmatrix{x\\y\\0}.$$
Now when talking about two vector spaces being the same, there are several philosophies. One is, you actually assume equality. In that sense, $\mathbb R^2$ is not a subspace of$\mathbb R^3$, as $\mathbb R^2$ consists of pairs of real numbers, and there are no pairs of real numbers in $\mathbb R^3$, only triples.
Another philosophy is to consider two vector spaces the same if they are the same up to isomorphism, that is, if they are isomorphic. And in that sense, $\mathbb R^2$ is clearly a subspace of $\mathbb R^3$, as shown above.
A third philosophy is to consider two vector spaces the same is there exists a canonical isomporphism between them, that is, not only are they isomorphic, but you can single out a specific isomporphism which is particularly simple, particularly obvious, or has particularly nice properties. The map given above is such a canonical embedding of $\mathbb R^2$ in $\mathbb R^3$, thus also in this sense, $\mathbb R^2$ is a subspace of $\mathbb R^3$.
A: Usually, an element $(x,y)$ of $\mathbb R^2$ is identified with $(x,y,0)\in\mathbb R^3$.
A: Some authors may say no because $(x,y)\in\Bbb R^2$ only has two coordinates;   whereas in $\Bbb R^3$ every point (vector)  has three coordinates. 
A: The only $\text{independent}$ subspaces of $\mathbb{R}^3$ are 
$(i)$ Lines passing through origin,
$(ii)$ planes passing through origin,
$(iii)$ $\mathbb{R}^3$, 
$(iv)$ $\{0 \}$, the zero subspace
Now look at $(i)$, any two mutually perpendicular lines through origin generates the plane $\mathbb{R}^2$.
Thus $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$.
