# Difference between Dimension of a Linear transformation (space) and the Dimension of its Column Space?

As my title suggests, what exactly is the difference?

The Column Space of a transformation $T: \mathbb R^n \to \mathbb R^m$ is simply the subspace which "contains" all the possible Images, right? If so, then what is the dimension of the matrix operator that defines the transformation? Why is it significant anyways? What information does it give us?

Any Help is Much Appreciated!
Thank You!

• I'm not familiar with the notion of "dimension of a transformation". Could you clarify this a bit? – Jason DeVito May 17 '12 at 19:54
• I mean the dimension of a matrix. Sorry – funktor May 18 '12 at 3:49

If that is the case, then if you express the transformation by left multiplication with a matrix $A$, then the dimension of the columnspace of $A$ is exactly the rank of $T$. The column vector you are operating on is merely making a linear combinations of the columns of $A$.