matrix spaces; rank 1; small world graphs
Problem 11.1: [Optional] (3.5 #41. Introduction to Linear Algebra: Strang) Write the 3 by 3 identity matrix as a combination of the other five permutation matrices. Then show that those five matrices are linearly independent. (Assume a combination gives c1P1 + ··· + c5P5 = 0 and check entries to prove ci is zero.) The five permutation matrices are a basis for the subspace of three by three matrices with row and column sums all equal.
Problem 11.2: (3.6 #31.) M is the space of three by three matrices. Multiply each matrix X in M by:
Notice that A .
- Which matrices X lead to AX = 0?
- Which matrices have the form AX for some matrix X?
- Part (a) finds the “nullspace” of the operation AX and part (b) finds the “column space.” What are the dimensions of those two subspaces of M? Why do the dimensions add to (n − r)+ r = 9?