# Monthly Archives: May 2012

## Linear Algebra and Its Applications, Exercise 2.4.7

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

## Linear Algebra and Its Applications, Exercise 2.4.6

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

## Linear Algebra and Its Applications, Exercise 2.4.5

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

## Linear Algebra and Its Applications, Exercise 2.4.4

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