线性代数习题详解(同济大学第四版第四章).pdf
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1v1 (1, 1, 0)T , v2 (0, 1, 1)T , v3 (3, 4, 0)T ,
v1 −v2 3v1 +2v2 −v3 .
T T
v1 −v2 (1, 1, 0) −(0, 1, 1)
(1 −0, 1 −1, 0 −1)T (1, 0, −1)T
3v +2v −v 3(1, 1, 0)T +2(0, 1, 1)T −(3, 4, 0)T
1 2 3
(3 ×1 +2 ×0 −3, 3 ×1 +2 ×1 −4, 3 ×0 +2 ×1 −0)T
(0, 1, 2)T
23(a1 −a) +2(a2 +a) 5(a3 +a) a1 (2,5,1,3)T ,
a2 (10,1,5,10)T , a3 (4,1,−1,1)T ,a
3(a1 −a) +2(a2 +a) 5(a3 +a)
1 1 T T T
a (3a +2a −5a ) [3(2,5,1,3) +2(10,1,5,10) −5(4,1,−1,1) ]
1 2 3
6 6
(1,2,3,4)T
3:
a , a , , a a a , a ,
(1) , .
1 2 m 1 2 m
(2)0 λ , λ , , λ
1 2 m
λa ++λ a +λb ++λ b 0
1 1 m m 1 1 m m
,a , , a , b , , b .
1 m 1 m
(3)λ , λ , , λ 0 ,
1 2 m
λa ++λ a +λb ++λ b 0
1 1 m m 1 1 m m
,a , , a , b , , b .
1 m 1 m
(4)a , , a , b , , b ,0 ,
1 m 1 m
λ , λ , , λ λa ++λ a 0, λb ++λ b 0
1 2 m 1 1 m m 1 1 m m
.
(1) a1 e1 (1,0,0, ,0)
a a a 0
2 3 m
a , a , , a a a , , a ,
, .
1 2 m 1 2 m
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