Below is a visualization of the coordinates for a subspace with respect to a given basis. [br][br]The ambient vector space is three dimensional. (The coordinate axes are hidden since they cluttered up the picture, but you can reveal them if you like by right clicked and selecting "show axes".) [br][br]The blue plane is a two dimensional subspace. It is given by x-y+z=0. Pick a basis, B, for this space by moving the blue points. Then move the Green point around on the plane to see how its actual corrdinates in the ambient space relate to its coordinates with respect to the basis B.

We haven't said anything about matrices yet. Form a matrix A whose columns are the vectors in the set B. Think of this matrix as a transformation. [br][br]What is its domain and range? [br][br]Is it one to one? Is it onto?[br][br]Can you describe what this transformation does?