# Monthly Archives: July 2012

## Linear Algebra and Its Applications, Exercise 2.4.16

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

## Linear Algebra and Its Applications, Exercise 2.4.15

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

## Linear Algebra and Its Applications, Exercise 2.4.14

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

## Linear Algebra and Its Applications, Exercise 2.4.13

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

## Linear Algebra and Its Applications, Exercise 2.4.12

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