8.7. EXERCISES 181

32. Consider the vectors of the form 2t +3s

s− tt + s

 : s, t ∈ R

 .

Is this set of vectors a subspace of R3? If so, explain why, give a basis for the sub-space and find its dimension.

33. Consider the vectors of the form

2t +3s+us− tt + s

u

 : s, t,u ∈ R

 .

Is this set of vectors a subspace of R4? If so, explain why, give a basis for the sub-space and find its dimension.

34. Consider the vectors of the form

2t +ut +3u

t + s+ vu

 : s, t,u,v ∈ R

 .

Is this set of vectors a subspace of R4? If so, explain why, give a basis for the sub-space and find its dimension.

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

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

37. Suppose A is an m×n matrix and {w1, · · · ,wk} is a linearly independent set of vec-tors in A(Fn)⊆Fm. Now suppose A(zi)=wi. Show {z1, · · · ,zk} is also independent.

38. 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.

39. 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.

40. 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.

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