证明显然 例6在R4中,求一单位向量与a1=(1,1,-1,1)2,a2=(1,-1,-1,1)2,a3=(2,1,1,3)均 正交 解设B=(x1,x2,x3,x4)与a1,a2,a3正交,则有 x1-x2-x3+x 1+x2+x3+3x4=0 解之,得到1=x4,22=0,x3=-4又叶+吗++=1解之,得到z4=土 所以 26 26’√26 习题 1.求证:对于欧氏空间V中的任意向量a,B,有 (1)|a+B2+la (2)(a,B)=a+2-la-B|2 2.(三角不等式)设V是欧氏空间,对于任意的a,B∈V,总有 a+≤lal+|l 3.在R4中,求a,B的夹角 (1)a=(2,1,3,2),B=(1 (2)a=(1,1,1,1),B=(0,1,0.0) 4.在欧氏空间V中,定义两个向量a,B的距离为d(a,B)=la-求证 (1)当a≠B时,d(a,B)>0; (3)d(a,B)≤d(a,)+d(,B) 5.在R4中,求与向量B=(1,-1,-1,1),7=(2,1,1,3)正交的所有向量 6.设51,52,…,5n是n维欧氏空间V的一个基,证明 (1)如果a∈V使得(a 0,1≤i≤m,那么a=0 (2)如果a1,a2∈V使得(a1,a)=(a2,a),1≤i≤n,那么a1=a2G: x\ ✷ 8 6  R 4 ￾Yt{H α1 = (1, 1, −1, 1)T , α2 = (1, −1, −1, 1)T , α3 = (2, 1, 1, 3)T > 5 4 β = (x1, x2, x3, x4) T  α1, α2, α3 5￾    x1 + x2 − x3 + x4 = 0 x1 − x2 − x3 + x4 = 0 2x1 + x2 + x3 + 3x4 = 0 9￾ x1 = 4 3 x4, x2 = 0, x3 = − 1 3 x4. Æ x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1, 9￾ x4 = ± √ 3 26 .  x4 = √ 3 26 d￾ x1 = √ 4 26 , x2 = 0, x3 = − √ 1 26 ;  x4 = − √ 3 26 d￾ x1 = − √ 4 26 , x2 = 0, x3 = √ 1 26 . l β = ±( 4 √ 26 , 0, − 1 √ 26 , 3 √ 26 ) T . CA 1. YVi0 V ℄{H α, β,  (1) |α + β| 2 + |α − β| 2 = 2|α| 2 + 2|β| 2 ; (2) (α, β) = 1 4 |α + β| 2 − 1 4 |α − β| 2 . 2. (`6g) V hVi0￾℄ α, β ∈ V , ! |α + β| ≤ |α| + |β|. 3.  R 4 ￾Y α, β /6 (1) α = (2, 1, 3, 2), β = (1, 2, −2, 1); (2) α = (1, 1, 1, 1), β = (0, 1, 0, 0). 4. Vi0 V ￾G{H α, β <Ar d(α, β) = |α − β|. Y (1)  α 6= β d￾ d(α, β) > 0; (2) d(α, β) = d(β, α); (3) d(α, β) ≤ d(α, γ) + d(γ, β). 5.  R 4 ￾Y{H β = (1, −1, −1, 1), γ = (2, 1, 1, 3) 5l{H 6. ξ1, ξ2, · · · , ξn h n sVi0 V *￾Q (1) ^" α ∈ V f (α, αi) = 0, 1 ≤ i ≤ n, SM α = 0; (2) ^" α1, α2 ∈ V f (α1, αi) = (α2, αi), 1 ≤ i ≤ n, SM α1 = α2. 3