8.7. EXERCISES 179
Show that this is a projection. Show that a vector in the column space of a projectionmatrix is left unchanged by multiplication by A.
9. Let H denote span
((12
),
(24
),
(13
)). Find the dimension of H and de-
termine a basis.
10. Let H denote span
1
20
,
240
,
131
,
011
. Find the dimension of
H and determine a basis.
11. Let H denote span
1
20
,
140
,
131
,
011
. Find the dimension of
H and determine a basis.
12. Let M ={
u = (u1,u2,u3,u4) ∈ R4 : u3 = u1 = 0}. Is M a subspace? Explain.
13. Let M ={
u = (u1,u2,u3,u4) ∈ R4 : u3 ≥ u1}. Is M a subspace? Explain.
14. Let w ∈ R4 and let M ={
u = (u1,u2,u3,u4) ∈ R4 : w ·u = 0}. Is M a subspace?
Explain.
15. Let M ={
u = (u1,u2,u3,u4) ∈ R4 : ui ≥ 0 for each i = 1,2,3,4}. Is M a subspace?
Explain.
16. Let w,w1 be given vectors in R4 and define
M ={
u = (u1,u2,u3,u4) ∈ R4 : w ·u = 0 and w1 ·u = 0}.
Is M a subspace? Explain.
17. Let M ={
u = (u1,u2,u3,u4) ∈ R4 : |u1| ≤ 4}. Is M a subspace? Explain.
18. Let M ={
u = (u1,u2,u3,u4) ∈ R4 : sin(u1) = 1}. Is M a subspace? Explain.
19. Study the definition of span. Explain what is meant by the span of a set of vectors.Include pictures.
20. Suppose {x1, · · · ,xk} is a set of vectors from Fn. Show that span(x1, · · · ,xk) contains0.
21. Study the definition of linear independence. Explain in your own words what ismeant by linear independence and linear dependence. Illustrate with pictures.
22. Use Corollary 8.5.18 to prove the following theorem: If A,B are n× n matrices andif AB = I, then BA = I and B = A−1. Hint: First note that if AB = I, then it must bethe case that A is onto. Explain why this requires span(columns of A) = Fn. Nowexplain why, using the corollary that this requires A to be one to one. Next explainwhy A(BA− I) = 0 and why the fact that A is one to one implies BA = I.