Difference Between Imagespace and Columnspace of a Matrix? I don't understand the difference between the columnspace of a matrix and the imagespace of a matrix. They are both the spanning sets of the columns of a matrix. Are they just different words for each other or is there a difference?
 A: The columnspace of a matrix is the vector space spanned by the columns of the matrix.
The image space a linear transformation is the set of all vectors that the transformation maps onto.
When you are interpreting the matrix as a linear transformation multiplying the left side of column vectors, then a column vector input combines the columns of the matrix. If you put in all possible inputs, you get all possible combinations of columns of the matrix. In this case, the columnspace is the same as the image space of the linear transformation.
But if you are using the matrix to multiply row vectors on the right instead, then the image of this transformation no longer matches the columnspace: it matches the rowspace! For now we will stick with the situation in the first paragraph, though.
I think a little more appreciation of the difference between transformations and matrices is called for. The image space of a transformation exists independently of whatever matrix you use to represent the transformation. No matter what matrix you compute for the transformation, the columns will always span the same space: the image space of the transformation.
