170 CHAPTER 9. SUBSPACES SPANS AND BASES

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

36. Suppose V and W both have dimension equal to 7 and they are subspaces of F10.What are the possibilities for the dimension of V ∩W? Hint: Remember that a linearindependent set can be extended to form a basis.

37. 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.

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

39. 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.

40. Suppose a system of linear equations has a 2×4 augmented matrix and the last col-umn is a pivot column. Could the system of linear equations be consistent? Explain.

41. 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.

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

43. 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 in thespan of four of the columns. Thus the columns are not independent.

(b) A is a 3×4 matrix, rank(A) = 3 and rank(A|b) = 2.

(c) A is a 4×2 matrix, rank(A) = 4 and rank(A|b) = 4. Hint: This says b is in thespan of the columns and the columns must be independent.

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

(e) A is a 4×2 matrix, rank(A) = 2 and rank(A|b) = 2.

44. Suppose A is an m×n matrix in which m≤ n. Suppose also that the rank of A equalsm. Show that A maps Fn onto Fm. Hint: The vectors e1, · · · ,em occur as columns inthe row reduced echelon form for A.

45. Suppose A is an m×n matrix in which m≥ n. Suppose also that the rank of A equalsn. Show that A is one to one. Hint: If not, there exists a vector x such that Ax= 0,and this implies at least one column of A is a linear combination of the others. Showthis would require the column rank to be less than n.

46. Explain why an n×n matrix A is both one to one and onto if and only if its rank is n.