second difference matrix

note about second difference matrix and dimension reduction

Clarification on the examples presented in Gilbert Strang. Introduction to linear algebra. Wellesley-Cambridge press, 5th ed edition. Chapter 10.2. Matrices in Engineering. There were some redundant columns in matrices displayed within the text without comment - I'm beating this issue to death here. The extra columns were permissible because of the special boundary conditions in the problem; and but the reasoning behind dropping a row and a column from the forward matrix is due to dimensionality reduction between the domain and co-domain of each difference matrix - not the boundary conditions which I thought was implied.