Kenneth Kuttler

168 CHAPTER 9. SUBSPACES SPANS AND BASES

16. Let M ={u= (u1,u2,u3,u4) ∈ R4 : |u1| ≤ 4

}. Is M a subspace? Explain.

17. Let M ={u= (u1,u2,u3,u4) ∈ R4 : sin(u1) = 1

}. Is M a subspace? Explain.

18. Suppose {x1, · · · ,xk} is a set of vectors from Fn. Show that span(x1, · · · ,xk) con-tains 0.

19. Prove the following theorem: If A,B are n× n matrices and if AB = I, then BA = Iand B = A−1. Hint: First note that if AB = I, then it must be the case that A is onto.Explain why this requires span(columns of A) = Fn. Now explain why, this requiresA to be one to one. Next explain why A(BA− I) = 0 and why the fact that A is oneto one implies BA = I.

20. Here are three vectors. Determine whether they are linearly independent or linearlydependent. (

1 2 0)T

2 0 1)T

3 0 0)T

21. Here are three vectors. Determine whether they are linearly independent or linearlydependent. (

4 2 0)T

2 2 1)T

0 2 2)T

22. Here are three vectors. Determine whether they are linearly independent or linearlydependent. (

1 2 3)T

4 5 1)T

3 1 0)T

23. Here are four vectors. Determine whether they span R3. Are these vectors linearlyindependent?(

1 2 3)T

4 3 3)T

3 1 0)T

2 4 6)T

24. Here are four vectors. Determine whether they span R3. Are these vectors linearlyindependent?(

1 2 3)T

4 3 3)T

3 2 0)T

2 4 6)T

25. 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

4 3 3)T

1 2 0)T

2 4 0)T

26. 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

16816.17.18.19.20.21.22.23.24.25.26.CHAPTER 9. SUBSPACES SPANS AND BASESLet M = {u = (u1,u2,u3,u4) € R*: |uj| <4}. Is M a subspace? Explain.Let M = {w= (u1,u2,u3,u4) € R*: sin(w;) = 1}. Is M a subspace? Explain.Suppose {2),--- ,a,} is a set of vectors from F”. Show that span (a,--- ,@) con-tains 0.Prove the following theorem: If A,B are n x n matrices and if AB = /, then BA =Iand B = A~!. Hint: First note that if AB = /, then it must be the case that A is onto.Explain why this requires span (columns of A) = F”. Now explain why, this requiresA to be one to one. Next explain why A (BA —/) = 0 and why the fact that A is oneto one implies BA = I.Here are three vectors. Determine whether they are linearly independent or linearlydependent.(1 2 0) (2 0 1)'.(3 0 0)Here are three vectors. Determine whether they are linearly independent or linearlydependent.(4 2 0)'.(2 2 1)'.(0 2 2)"Here are three vectors. Determine whether they are linearly independent or linearlydependent.T T T(1 2 3) (4 5 1) .(3 1 0 )Here are four vectors. Determine whether they span R?. Are these vectors linearlyindependent?(1 2 3) (4 3 3)'.(3 1 0) (2 4 6)Here are four vectors. Determine whether they span R*. Are these vectors linearlyindependent?(1 2 3)'(4 3 3)'.(3 2 0) (2 4 6)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) (4 3 3)(1 2 0) (2 4 0)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 0) (1 2 0)