Monthly Archives: September 2011

Exercise 2.3.23. Let through be vectors in . Answer the following questions: a) Are the nine vectors linearly independent? Not linearly independent? Might be linearly independent? b) Do the nine vectors span ? Not span ? Might span ? c) … Continue reading →

Exercise 2.3.22. Given a vector space of dimension 7 and a subspace of of dimension 4, state whether the following are true or false: 1) You can create a basis for by adding three vectors to any set of vectors … Continue reading →

Exercise 2.3.21. Suppose is a 64 by 17 matrix and has rank 11. How many independent vectors are solutions to the system ? What about the system ? Answer: If the rank of is 11 then performing elimination on produces … Continue reading →

Exercise 2.3.20.Consider the set of all 2 by 2 matrices that have the sum of their rows equal to the sum of their columns. What is a basis for this subspace? Consider the analogous set of 3 by 3 matrices … Continue reading →

Exercise 2.3.19. Suppose is an by matrix, with columns taken from . What is the rank of if its column vectors are linearly independent? What is the rank of if its column vectors span ? What is the rank of … Continue reading →

Exercise 2.3.18. Indicate whether the following statements are true or false: a) given a matrix whose columns are linearly independent, the system has one and only solution for any right-hand side b) if is a 5 by 7 matrix then … Continue reading →

Exercise 2.3.17. Suppose that and are subspaces of , each with dimension 3. Show that and must have at least one vector in common other than the zero vector. Answer: Since and each have dimension 3 their respective bases each … Continue reading →

Exercise 2.3.16. What is the dimension of the vector space consisting of all 3 by 3 symmetric matrices? What is a basis for it? Answer: There are nine possible entries that can be set in a 3 b 3 matrix, … Continue reading →

Exercise 2.3.15. If the vector space has dimension show that a) if a set of vectors in is linearly independent then that set forms a basis b) if a set of vectors in spans then that set forms a basis … Continue reading →

Exercise 2.3.14. Suppose we have the following matrix How can you extend the rows of to create a basis for ? How can you reduce the columns of to create a basis for ? Answer: As defined is in echelon … Continue reading →