112 CHAPTER 4. LINEAR SPACES
Also, since (u− z,v) = 0 for all v ∈V, |u|2 = |u− z+ z|2 = |u− z|2 + |z|2 so
|u|2 =
∣∣∣∣∣u− n
∑i=1
zivi
∣∣∣∣∣2
+
∣∣∣∣∣ n
∑i=1
zivi
∣∣∣∣∣2
= d2 +
∣∣∣∣∣ n
∑i=1
zivi
∣∣∣∣∣2
= d2 +∑j
=(u,v j)︷ ︸︸ ︷∑
i(v j,vi)ziz j = d2 +∑
j(u,v j)z j
= d2 +yTz, y ≡ ((u,v1) , · · · ,(u,vn))T , z ≡
(z1, · · · ,zn)T
From ∗, Gz = y,
(G(v1, ...,vn) 0
yT 1
)(zd2
)=
(y
∥u∥2
). Now use Cramer’s
rule to solve for d2 and get
d2 =
det(
G(v1, ...,vn) y
yT |u|2)
det(G(v1, ...,vn))≡ detG(v1, ...,vn,u)
detG(v1, ...,vn)
23. In the situation of Problem 21, let fk (x)≡ xk and let V ≡ span( fp1 , ..., fpn). give anestimate for the distance d between fm and V for m a nonnegative integer and as inthe above problem − 1
2 < p1 < · · · < pn. Use Theorem 1.9.28 in the appendix andthe above problem with vi ≡ fpi and vn+1 ≡ fm. Justify the following manipulations.
The numerator in the above formula for the distance is of the form ∏ j<i≤n+1(pi−p j)2
∏i, j≤n+1(pi+p j+1)
=∏ j<i≤n (pi− p j)
2∏ j≤n (m− p j)
2
∏i, j≤n (pi + p j +1)∏ni=1 (pi +m+1)∏
nj=1 (p j +m+1)(2m+1)
While G( fp1 , ..., fpn) =∏ j<i≤n(pi−p j)
2
∏i, j≤n(pi+p j+1). Thus d =
∏ j≤n|m−p j|∏
ni=1(pi+m+1)(
√2m+1)
.
24. Suppose ∑nk=0 aktk = 0 for each t ∈ (−δ ,δ ) where ak ∈ X , a linear space. Show that
each ak = 0.
25. Suppose A ⊆ Rp is covered by a finite collection of Balls F . Show that then thereexists a disjoint collection of these balls, {Bi}m
i=1, such that A⊆∪mi=1B̂i where B̂i has
the same center as Bi but 3 times the radius. Hint: Since the collection of balls isfinite, they can be arranged in order of decreasing radius. Mimic the argument forVitali covering theorem.
112 CHAPTER 4. LINEAR SPACESAlso, since (u—z,v) =0 for all v € V, |u|? = |u—z+z)? = |u—z|? + |z|” so2 2n n n|u|? = u—\z'v; + Vivi = d+ Vivii=l i=l i=l=(uv/)5 a=P + PV vj viziz! =a? +) (u,vj) 2!joi jT=d+4+y'z, y= ((u,v1) 50° .(U,Vn))’ , Z= (z! ++ iz")G(v1,..-5¥n) O Zz yFrom *, Gz = = . Now use Cramer’s( yo JAP |?rule to solve for d? and getaet( GMs --5¥n) y, )pe y |u| _ detG(v1,...,Vn,W)7 det (G(v1,.-.,Vn)) ~ detG(v1,...;¥n)23. In the situation of Problem 21, let f; (x) = x* and let V = span (fpys-0 tpn)» give anestimate for the distance d between f,, and V for m a nonnegative integer and as inthe above problem —5 < pi <+++< py. Use Theorem 1.9.28 in the appendix andthe above problem with v; = f,, and vn41 = fm. Justify the following manipulations.2The numerator in the above formula for the distance is of the form Mjcisnt (Pipi)Tlij<n+1 (pitpj+ 1)2 2_ Ilj<i<n (Pi- Pj) Tj<n (m— pj)Tij<n (Pit Pi t+) (Vit m+ 1) Tj) (pj +m 1) (2m+ 1): _ Ij<i<n (pi-p;)° _ Ti<n|m—p,|While G(fp, ’ st pn) a Ti j<n (pit pj+1) - Thus d = " \(pitm+1)(V2m+1) .24. Suppose Y"_p axt* = 0 for each t € (— 6,5) where a, € X, a linear space. Show thateach a, = 0.25. Suppose A C R? is covered by a finite collection of Balls . Show that then thereexists a disjoint collection of these balls, {Bi} ie , such that A C U"_Bi where B; hasthe same center as B; but 3 times the radius. Hint: Since the collection of balls isfinite, they can be arranged in order of decreasing radius. Mimic the argument forVitali covering theorem.