Algebra And Geometry Combined Explains How The Mind Does Math

This paper updates my talk on Cache Blocking for Dense Linear Algorithms since 1985 given at PPAM 11; see [11]. We again apply Dimension Theory to matrices in the Fortran and C programming languages. New Data Structures (NDS) for matrices are given. We use the GCD algorithm to transpose a n by m matrix A in CMO order, standard layout, in-place. Algebra and Geometry are used to make this idea concrete and practical; it is the reason for title of our paper: make a picture of any matrix by the GCD algorithm to convert it into direct sum of square submatrices. The picture is Geometry and the GCD algorithm is Algebra. Also, the in-place transposition of the GKK and TT algorithms will be compared. Finally, the importance of using negative integers will be used to give new results about subtraction and finding primitive roots which also make a priori in-place transpose more efficient.