18.6. EXERCISES 405
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 Make them the columns
of a matrix and row reduce to determine whether they are linearly independent.
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 if they span R3. Are these vectors linearly inde-pendent?(
1 2 3)T
,(
4 3 3)T
,(
3 1 0)T
,(
2 4 6)T
24. Here are four vectors. Determine if they span R3. Are these vectors linearly inde-pendent?
(1 2 3
)T,(
4 3 3)T
,(
3 2 0)T
,(
2 4 6)T
25. Determine if the following vectors are a basis for R3. If they are, explain why theyare 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. 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.
27. 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.