TITLE: Four Fundamental subspaces
QUESTION [5 upvotes]: Given an $m \times n$ matrix, where $m$ is the number of rows and $n$ is the number of columns. 

Four Fundamental Subspaces

The row space is $C(A^t)$, a subspace of $\mathbb{R}^n$.
The column space is $C(A)$, a subspace of $\mathbb{R}^m$.
The nullspace is $N(A)$, a subspace of $\mathbb{R}^n$.
The left nullspace is $N(A^t)$, a subspace of $\mathbb{R}^m$. This is our new space.


If the column space is the space that is spanned by the column vectors, why is it that it is a subspace of $\mathbb{R}^m$ and not $\mathbb{R}^n$, since the dimension of the columns should be $n$ instead? 
Or am I missing something fundamental here?

REPLY [2 votes]: Since the dimension of the columns should n instead?

No, you're misinterpreting the dimensions of a matrix. Given an $m\times n$ matrix (I'm going to stick with lowercase letters), then as you said correctly yourself,

M is number of rows and N is number of columns.

So $n$ is the number of columns, but not their dimension (or size). Each column has $m$ entries in it because there are $m$ rows in the matrix, i.e. $m$ is the "height" of the matrix and thus the "height" of each column. That's why each column is an element of $\mathbb{R}^m$.