On 26/04/2015, at 1:53 am, Mike Meyer
My real problem is that I've got a list of points in R3 and want to decide if they determine a plane, meaning they are coplanar but not colinear. Similarly, given a list of points in R2, I want to verify that they aren't colinear. Both of these can be done by converting the list of points to a matrix and finding the rank of the matrix, but I only use the rank function in the definitions of colinear and coplanar.
To compute the rank of a matrix, perform elementary row operations until the matrix is left in echelon form; the number of nonzero rows remaining in the reduced matrix is the rank. (http://www.cliffsnotes.com/math/algebra/linear-algebra/real-euclidean-vector...) A matrix is in row echelon form when it satisfies the following conditions: * The first non-zero element in each row, called the leading entry, is 1 * Each leading entry is in a column to the right of the leading entry in the previous row * Rows with all zero elements, if any, are below rows having a non-zero element. (http://stattrek.com/matrix-algebra/echelon-transform.aspx) Row echelon forms aren't unique, but for determining the rank of a matrix, that doesn't matter. Code working on a list of points left as an exercise for the reader.