sentences:  Intuitively, given a matrix "A", the action of the matrix "A" on a vector x will return a linear combination of the columns of "A" weighted by the coordinates of x as coefficients.Another way to look at this is that it will (1) first project x into the row space of "A", (2) perform an invertible transformation, and (3) place the resulting vector y in the column space of "A".Thus the result y = "A" x must reside in the column space of "A".