Pictorial A Representation of the Forward Problem
Gidon Eshel
This schematic shows the actions associated with Ax = b, where
A RM N x RN b RM .
The upper (left) panel is a 3D schematic representation of the vector space that contains x, RN . The vector x on which A operates (i.e., which A pre-multiplies) is shown in magenta. It comprises 2 mutually orthogonal parts (shown in cyan and blue, respectively), x = xR(AT ) + xN (A) that reside in 2 mutually orthogonal sub-spaces of RN , As row space R(AT ) (the red z = 0 x y-plane), and As nullspace N (A) (pictorially, the region above the red plane, the entire space except the z = 0 plane). (Recall that the row space has the same dimension as As rank, and it comprises all vectors that can be represented as a linear combination of As rows. One reasonable spanning set for R(A) is the rows with non-zero pivots remain that after reducing A to U using elementary row operations.) By denition, AxN (A) = 0, which is shown by the black arrow from the tip of xN (A) to the origin of the lower (right) space, the schematic representation of RM , the vector space containing b. The remainder of x, xR(AT ) , is mapped by A onto As column space R(A), schematically shown in the gure as the pinkish-red z = 0 plane in the lower (right) space representing RM . If we were simply going from RN to RM by carrying out Ax and thus forming b = ...