Consider the 10x10 matrix M defined by this file "m10x10.mat":

10 10

2 1 0 0 0 0 0 0 0 0

1 2 1 0 0 0 0 0 0 0

0 1 2 1 0 0 0 0 0 0

0 0 1 2 1 0 0 0 0 0

0 0 0 1 2 1 0 0 0 0

0 0 0 0 1 2 1 0 0 0

0 0 0 0 0 1 2 1 0 0

0 0 0 0 0 0 1 2 1 0

0 0 0 0 0 0 0 1 2 1

0 0 0 0 0 0 0 0 1 2

Note: This matrix is symmetric, making it "sparse" in the sense that most entries are zero.

Consider also the dimension 10 vector V defined by this file "v10.vec": 10 0 0 0 0 1 1 0 0 0 0 Compute the following items:

The vector W = MV

The sparse matrix representation m of M

The sparse vector representation v of V

The sparse vector representation w of W

The sparse product x = mv

Then verify that the sparse vectors w and x are equal.