Kenneth Kuttler

9.2. EXERCISES 169

27. Determine whether the following vectors are a basis for R3. If they are, explain whythey are and if they are not, give a reason and tell whether they span R3.(

1 0 3)T

0 1 0)T

1 2 0)T

0 0 0)T

28. Determine whether the following vectors are a basis for R3. If they are, explain whythey are and if they are not, give a reason and tell whether they span R3.(

1 0 3)T

0 1 0)T

1 1 3)T

0 0 0)T

29. 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.

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

2t +3s+us− tt + s

 : 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.

31. 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.

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

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

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

9.2. EXERCISES 16927.28.29.30.31.32.33,34.Determine whether the following vectors are a basis for R*. If they are, explain whythey are and if they are not, give a reason and tell whether they span R?.(1 0 3)'.(0 1 0) (1 2 0) (0 0 0)Determine whether the following vectors are a basis for >. If they are, explain whythey are and if they are not, give a reason and tell whether they span R?.(1 0 3).(0 i 0) (1 1 3)'.(0 0 0)Consider the vectors of the form2t+3ss—t :s,tERt+sIs this set of vectors a subspace of R?? If so, explain why, give a basis for the sub-space and find its dimension.Consider the vectors of the form2t+3s+us—tt+su:s,t,uERIs this set of vectors a subspace of R*? If so, explain why, give a basis for the sub-space and find its dimension.Consider the vectors of the form2t+ut+3ut+s+vu:s,t,u,vERIs this set of vectors a subspace of R*? If so, explain why, give a basis for the sub-space and find its dimension.If you have 5 vectors in FF and the vectors are linearly independent, can it always beconcluded they span F>? Explain.If you have 6 vectors in F°, is it possible they are linearly independent? Explain.Suppose A is an m xn matrix and {wy),---,w,;} is a linearly independent set ofvectors in A (F”) C F”. Now suppose A (z;) = w;. Show {z1,--- ,2;} is also inde-pendent.