What do we call the result of wedging together the columns of a matrix? We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times a$ submatrix. This can also be used to test linear dependence among the columns, and to compute the rank.

Question. The result of wedging together the columns of a matrix $A$ in the order they appear is called the [what] of $A$?

I'm also interested in answers of the form: "Really, this should be thought of as something you can do to a linear transform, and its called the [whatever] of a linear transform."
 A: At the risk of oversimplifying things, this has all the information of the image.
When you wedge together $k$ vectors, you get a description of the $k$-dimensional subspace spanned by those vectors--in particular, the components of such a wedge product are coefficients used in a linear combination of basis $k$-dimensional subspaces.
When this process is applied to a $b\times a$ matrix $A$, what is that subspace described?  If the subspace is nondegenerate, then this is identical to the image subspace of $A$.  If no more than $k < b$ vectors are linearly independent, then the wedge product of those $k$ such vectors describes the image of $A$.
Other terms for this include column space, range, etc.
However, the wedge product will have a scale factor that the image (etc.) does not describe.  In the $a \times a$ case, this is the determinant, describing how a unit $a$-volume is magnified or shrunk by the transformation.  The same logic applies here: the scale factor describes how an $a$-volume in the domain is magnified or shrunk into the image $b$-volume. There is also information about the change of orientation.
