Category Archives: linear algebra

Exercise 2.5.2. Given the incidence matrix from exercise 2.5.1 and any vector in the column space of show that . Prove the same result based on the rows of . What is the implication for the potential differences around a … Continue reading →

Exercise 2.5.1. Describe the incidence matrix for the following graph: The graph has three nodes and three edges, with edge 1 going from node 2 to node 1, edge 2 going from node 3 to node 2, and edge 3 … Continue reading →

Exercise 2.4.21. Suppose that for two matrices and the associated subspaces (column space, row space, null space, and left nullspace) are the same. Does this imply that ? Answer: The answer is no, as shown by the following counterexample: We … Continue reading →

Exercise 2.4.20. For each of the following properties find a matrix with that property. If no such matrix exists, explain why that is the case. a) The column space of the matrix contains the vectors and the row space contains … Continue reading →

Exercise 2.4.19. For the following matrix find a basis for each of the four associated subspaces. Answer: Per the above equation the matrix on the left side can be factored into a lower triangular matrix with unit diagonal and an … Continue reading →

Exercise 2.4.18. Given the vectors and the subspace that they span, find two matrices and such that . Answer: The easiest way to create a matrix whose row space is is to use the vectors above as the rows of … Continue reading →

Exercise 2.4.17. Point out the error in the following argument: Suppose is a right-inverse of so that . Then we can multiply both sides by to produce or . But we then have so that is a left-inverse as well. … Continue reading →

Exercise 2.4.16. Given an by matrix the columns of which are linearly independent, fill in the blanks in the following statements: The rank of is ____. The nullspace is ____. The row space is ____. There is at least one … Continue reading →

Exercise 2.4.15. For each of the following matrices find a left inverse and right inverse if they exist. Answer: We begin with the 2 by 3 echelon matrix . Since has two pivots its rank . Since has two rows … Continue reading →

Exercise 2.4.14. Suppose we have the following matrix: with , , and given and . For what value of does have rank 1? In this case how can be expressed as the product of a column vector and row vector … Continue reading →