Category Archives: linear algebra

Exercise 2.4.13. What is the rank of each of the following matrices: Express each matrix as a product of a column vector and row vector, . Answer: We do Gaussian elimination on the first matrix by subtracting two times the … Continue reading →

Exercise 2.4.12. Suppose that for a matrix the system has at least one nonzero solution. Show that there exists at least one vector for which the system has no solution. Show an example of such a matrix and vector . … Continue reading →

Exercise 2.4.11. Suppose that for a matrix and any the system always has at least one solution. Show that in this case the system has no solution other than . Answer: Suppose is an by matrix and is an -element … Continue reading →

Exercise 2.4.10. Suppose the nullspace of a matrix is the set of all vectors in for which . Find a 1 by 3 matrix with this nullspace. Find a 3 by 3 matrix with the same nullspace. Answer: If and … Continue reading →

Exercise 2.4.9. Let be a matrix representing a system of equations in unknowns, and assume that the only solution to is 0. What is the rank of ? Explain your answer. Answer: is a linear combination of the columns of … Continue reading →

Exercise 2.4.8. Is it possible for the row space and nullspace of a matrix to both contain the vector ? If not, why not? Answer: Suppose is an by 3 matrix and is in the nullspace . Then we must … Continue reading →

Exercise 2.4.7. If is an by matrix with rank answer the following: a) When is invertible, with existing such that ? b) When does have an infinite number of solutions for any ? Answer: a) Per theorem 2Q on page … Continue reading →

Exercise 2.4.6. Given a matrix with rank show that the system has a solution if and only if the matrix also has rank , where is formed by taking the columns of and adding as an additional column. Answer: We … Continue reading →

Exercise 2.4.5. Suppose that for two matrices and . Show that the column space is contained within the nullspace and that the row space is contained within the left nullspace . Answer: Assume that is an by matrix and is … Continue reading →

Exercise 2.4.4. For the matrix describe each of its four associated subspaces. Answer: We first consider the column space . The matrix has two pivots (in the second and third columns) and therefore rank ; this is the dimension of … Continue reading →