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!

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). 
