1. Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: C 0 = C 1 = C 2 = C 3 = Z 0 = Explain your answer for Z 0 : B 0 = H 0 = Rank C 0 = Rank Z 0 = Rank B 0 = Rank H 0 = |C 0 | = |Z 0 | = |B 0 | = |H 0 |= v4v4 v3v3 v1v1 v2v2

2. Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: Find the matrix for  0 : Find the matrix for  1 : Simplify the matrix for  1 so that the nonzero columns are linearly independent. Write the basis element above its corresponding column. v4v4 v3v3 v1v1 v2v2

3. Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: Find the matrix for  2 : Simplify the matrix for  2 so that the nonzero columns are linearly independent. Write the basis element above its corresponding column. v4v4 v3v3 v1v1 v2v2

4. Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: Z 1 = Explain your answer for Z 1 : B 1 = H 1 = Rank C 1 = Rank Z 1 = Rank B 1 = Rank H 1 = |C 1 | = |Z 1 | = |B 1 | = |H 1 |= v4v4 v3v3 v1v1 v2v2

5. Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: Z 2 = Explain your answer for Z 2 : B 2 = H 2 = Rank C 2 = Rank Z 2 = Rank B 2 = Rank H 2 = |C 2 | = |Z 2 | = |B 2 | = |H 2 |= v4v4 v3v3 v1v1 v2v2

6. Barcode for H 0 Barcode for H 1