210 CHAPTER 7. VECTOR SPACES AND FIELDS

14. Let V denote the set of polynomial functions defined on [0, 1]. Vector addition isdefined as (f + g) (x) ≡ f (x)+g (x) and scalar multiplication is defined as (αf) (x) ≡α (f (x)). Verify V is a vector space. What is its dimension, finite or infinite? Justifyyour answer.

15. Let V be the set of polynomials defined on R having degree no more than 4. Give abasis for this vector space.

16. Let the vectors be of the form a + b√2 where a, b are rational numbers and let the

field of scalars be F = Q, the rational numbers. Show directly this is a vector space.What is its dimension? What is a basis for this vector space?

17. Let V be a vector space with field of scalars F and suppose {v1, · · · ,vn} is a basis forV . Now let W also be a vector space with field of scalars F. Let L : {v1, · · · ,vn} →W be a function such that Lvj = wj . Explain how L can be extended to a lineartransformation mapping V to W in a unique way.

18. If you have 5 vectors in F5 and the vectors are linearly independent, can it always beconcluded they span F5? Explain.

19. If you have 6 vectors in F5, is it possible they are linearly independent? Explain.

20. Suppose V,W are subspaces of Fn. Show V ∩W defined to be all vectors which are inboth V and W is a subspace also.

21. Suppose V and W both have dimension equal to 7 and they are subspaces of a vectorspace of dimension 10. What are the possibilities for the dimension of V ∩W? Hint:Remember that a linear independent set can be extended to form a basis.

22. Suppose V has dimension p and W has dimension q and they are each contained ina subspace, U which has dimension equal to n where n > max (p, q) . What are thepossibilities for the dimension of V ∩W? Hint: Remember that a linear independentset can be extended to form a basis.

23. If b ̸= 0, can the solution set of Ax = b be a plane through the origin? Explain.

24. Suppose a system of equations has fewer equations than variables and you have founda solution to this system of equations. Is it possible that your solution is the only one?Explain.

25. Suppose a system of linear equations has a 2×4 augmented matrix and the last columnis a pivot column. Could the system of linear equations be consistent? Explain.

26. Suppose the coefficient matrix of a system of n equations with n variables has theproperty that every column is a pivot column. Does it follow that the system ofequations must have a solution? If so, must the solution be unique? Explain.

27. Suppose there is a unique solution to a system of linear equations. What must be trueof the pivot columns in the augmented matrix.

28. State whether each of the following sets of data are possible for the matrix equationAx = b. If possible, describe the solution set. That is, tell whether there exists aunique solution no solution or infinitely many solutions.

(a) A is a 5 × 6 matrix, rank (A) = 4 and rank (A|b) = 4. Hint: This says b is inthe span of four of the columns. Thus the columns are not independent.