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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!

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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
up vote 2 down vote accepted

It sounds to me like you mean "rank of a transformation" as in the dimension of the transformation's image.

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 $A$.

There are several terms floating around here that are almost the same.

  • rank of a linear transformation (there is no such thing as "the" columnspace or rowspace of a transformation, but the dimension of the image is well-defined.)
  • column rank of a matrix (dimension of columnspace of the matrix)
  • row rank of a matrix (dimension of the rowspace of a matrix)

I think you will be able to sort it all out from this wiki article.

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