Category Archives: linear algebra

Exercise 2.5.12. Let be a 12 by 7 incidence matrix for a connected graph. What are the following values? the rank of the number of free variables in the solution for the system the number of free variables in the … Continue reading →

Exercise 2.5.11. Given the incidence matrix from exercise 2.5.10 with the final column removed and a diagonal matrix with elements 1, 2, 2, and 1, write the equations for the system Show that eliminating gives the system and solve the … Continue reading →

Exercise 2.5.10. Given the incidence matrix draw the graph corresponding to the matrix, and state whether or not it is a tree and the rows are linearly independent. Demonstrate that removing a row produces a spanning tree, and describe the … Continue reading →

Exercise 2.5.9. Given the incidence matrix from exercise 2.5.6 (for the graph on page 113 with six edges and four nodes) and the diagonal matrix compute and . Describe how the diagonal and other entries in can be predicted from … Continue reading →

Exercise 2.5.8. State the dimensions of the four fundamental subspaces of the incidence matrix from exercise 2.5.6 (for the graph on page 113 with six edges and four nodes) and provide a set of basis vectors for each subspace. Answer: … Continue reading →

Exercise 2.5.7. Suppose that the incidence matrix from exercise 2.5.6 (for the graph on page 113 with six edges and four nodes) represents six games among four teams, and that the score differences for the six games are through . … Continue reading →

Exercise 2.5.6. Determine the incidence matrix for the graph on page 113 with six edges and four nodes, as follows: Edge 1 from node 1 to node 2, edge 2 from node 1 to node 3, edge 3 from node … Continue reading →

Exercise 2.5.5. Given the incidence matrix from exercise 2.5.1 and the diagonal matrix compute and show that the 2 by 2 matrix resulting from removing the third row and third column is invertible.. Answer: From exercise 2.5.1 we have the … Continue reading →

Exercise 2.5.4. Given the incidence matrix from exercise 2.5.1 show that is symmetric and singular, and determine its nullspace. Show that the matrix obtained by removing the last row and column of is nonsingular. Answer: From exercise 2.5.1 we have … Continue reading →

Exercise 2.5.3. Given the incidence matrix from exercise 2.5.1 and any vector in the row space of show that . Prove the same result based on the linear system . What is the implication if , , and are currents … Continue reading →