Question about Isomorphism I was reading this example on Wikipedia.
It says the following:
Another example is the quotient of $\mathbb{R}^n$ by the subspace spanned by the first $m$ standard basis vectors. The space $\mathbb{R}^n$ consists of all $n$-tuples of real numbers $(x_1, ..., x_n)$. The subspace, identified with $\mathbb{R}^m$, consists of all $n$-tuples such that the last $n-m$ entries are zero: $(x_1, ..., x_m, 0, 0, ..., 0)$. Two vectors of $\mathbb{R}^n$ are in the same congruence class modulo the subspace if and only if they are identical in the last $n-m$ coordinates. The quotient space $\mathbb{R}^n/\mathbb{R}^m$ is isomorphic to $\mathbb{R}^{n-m}$ in an obvious manner.
I don't understand the last line. Why is $\mathbb{R}^n/\mathbb{R}^m$ is isomorphic to $\mathbb{R}^{n-m}$? Can anyone please help? Maybe I am not clear on what is an isomorphism. Can anyone explain?
Thanks!
 A: Because the last $m$ coordinates are all $0$, so the subspace has dimension n-m and so it is isomorphic to $\mathbb R^{n-m}$. The isomorphism send $(v_1,...,v_{n-m},0_{n-m+1},...,0_n)\in \mathbb R^{n}$ in $(v_1,...,v_{n-m})\in\mathbb R^{n-m}$. You can verify that this is an isomorphism (that is a linear map injective and surjective) by proving that it's injective and surjective (obvious by the dimension of your subspace). 
A: You can write the quotient space as $\mathbb{R}^n/\mathbb{R}^m$={v + $\mathbb{R}^m$} with v $\in$ $\mathbb{R}^n$.  When we consider the projection p from $\mathbb{R}^n$ to $\mathbb{R}^n/\mathbb{R}^m$, we can see that its kernel is: ker(p)={v $\in$ $\mathbb{R}^m$}, since p(v)=0 for all v $\in$  $\mathbb{R}^m$. Therefore, it the dimension of $\mathbb{R}^n/\mathbb{R}^m$ has to be n-m, since dim(ker(p))+dim(im(p))=n and dim(ker(p))=m.
A: Another way you can think of it is that: the projection $\mathbb{R}^n\to\mathbb{R}^{n-m}$, given by $(a_1,a_2,...,a_{n-m},a_{n-m+1},...,a_n)\mapsto (a_1,a_2,...,a_{n-m},0,...,0)$ is a linear map whose kernel is $\mathbb{R}^m$. 
Then, the isomorphism $\mathbb{R}^n/\mathbb{R}^m\cong\mathbb{R}^{n-m}$ is just the first isomorphism theorem for vector spaces (and linear maps). 
