Monthly Archives: November 2013

Review exercise 2.17. Suppose that is a vector in and that for all . Show that must be the zero vector. Answer: For and we have Since  for all we must have for each of the elementary vectors , , … Continue reading →

Review exercise 2.16. For each of the cases in review exercise 2.15, describe the relationship among the rank of , the number of rows , and the number of columns . Answer: (i) For the matrix we have , , … Continue reading →

Review exercise 2.15. For each of the following, find a matrix for which i) there is either no solution or one solution to depending on ii) there are an infinite number of solutions to for all iii) there is either … Continue reading →

Review exercise 2.14. Do the three vectors , , and form a basis for the vector space ? Answer: In order for the three vectors to be a basis for they must be linearly independent. We can test this by … Continue reading →

Review exercise 2.13. For the matrix find its triangular factors and describe the conditions under which the columns of are linearly independent. Answer: We start elimination by subtracting 1 times the first row from the second row, with We next … Continue reading →

Review exercise 2.12. The matrix is by and has rank . What is the dimension of its nullspace? Answer: The dimension of the nullspace is the number of columns of minus the rank of , or . NOTE: This continues … Continue reading →

Review exercise 2.11. a) Given the following matrix find its rank and a basis for the nullspace. b) Are the first three rows of a basis for the row space of ? Are the first, third, and sixth columns of … Continue reading →

Review exercise 2.10. Given the set of all linear transformations from to define operations for scalar multiplication and vector addition that will make the set a vector space. What is the dimension of the resulting vector space? Answer: Let and … Continue reading →