题目

设向量组(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T不能由向量组(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T线性表示.（1）求a的值；（2）将(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T用(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T，(alpha )_(1)=((1,0,1))^T线性表示.

设向量组 $%5Calpha%20_%7B1%7D%3D%5Cleft%281%2C0%2C1%5Cright%29%5E%7BT%7D$ ， $%5Calpha%20_%7B2%7D%3D%5Cleft%280%2C1%2C1%5Cright%29%5E%7BT%7D$ ， $%5Calpha%20_%7B3%7D%3D%5Cleft%281%2C3%2C5%5Cright%29%5E%7BT%7D$ 不能由向量组 $%5Cbeta%20_%7B1%7D%3D%5Cleft%281%2C1%2C1%5Cright%29%5E%7BT%7D$ ， $%5Cbeta%20_%7B2%7D%3D%5Cleft%281%2C2%2C3%5Cright%29%5E%7BT%7D$ ， $%5Cbeta%20_%7B3%7D%3D%5Cleft%283%2C4%2Ca%5Cright%29%5E%7BT%7D$ 线性表示.

（1）求a的值；

（2）将 $%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D$ 用 $%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D$ 线性表示.

题目解答

答案

（1）由于 $%5Calpha%20_%7B1%7D%3D%5Cleft%281%2C0%2C1%5Cright%29%5E%7BT%7D$ ， $%5Calpha%20_%7B2%7D%3D%5Cleft%280%2C1%2C1%5Cright%29%5E%7BT%7D$ ， $%5Calpha%20_%7B3%7D%3D%5Cleft%281%2C3%2C5%5Cright%29%5E%7BT%7D$ 不能由 $%5Cbeta%20_%7B1%7D%3D%5Cleft%281%2C1%2C1%5Cright%29%5E%7BT%7D$ ， $%5Cbeta%20_%7B2%7D%3D%5Cleft%281%2C2%2C3%5Cright%29%5E%7BT%7D$ ， $%5Cbeta%20_%7B3%7D%3D%5Cleft%283%2C4%2Ca%5Cright%29%5E%7BT%7D$ 线性表出，所以 $%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D$ 线性相关（因为任意n+1个n维向量线性相关，从而 $%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D$ ， $%5Calpha%20_%7Bi%7D%5Cleft%28i%3D1%2C2%2C3%5Cright%29$ 线性相关，若 $%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D$ 线性无关，则 $%5Calpha%20_%7Bi%7D$ 可由 $%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D$ 线性表示，从而 $%7C%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D%7C%3D0$ ，而 $%7C%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D%7C%3D.%201%201%203%201%202%204%201%203%20a%20.%3D.%201%201%203%200%201%201%200%202%20a-3%20.%3Da-5$ ，故可解得a=5

（2）设 $%28%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D%29%3D%28%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D%29A$ ，由于 $%7C%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D%7C%3D.%201%200%201%200%201%203%201%201%205%20.%3D1%5Cneq%200$ ，所以 $%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D$ 线性无关.则 $A%3D%28%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D%29%5E%7B-1%7D%28%5Cbeta%20_%7B1%7D$ ， $%5Cbeta%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D%29$

而 $%28%5Calpha%20_%7B1%7D$ ， $%5Calpha%20_%7B2%7D$ ， $%5Calpha%20_%7B3%7D%29%5E%7B-1%7D%3D%202%201%20-1%203%204%20-3%20-1%20-1%201%20$ ，从而A= 2 1 -1 3 4 -3 -1 -1 1 1 1 3 1 2 4 1 3 5 = 2 1 5 4 2 10 -1 0 -2

因此 $%5Cbeta%20_%7B1%7D%3D2%5Calpha%20_%7B1%7D%2B4%5Calpha%20_%7B2%7D-%5Calpha%20_%7B3%7D$ ， $%5Cbeta%20_%7B2%7D%3D%5Calpha%20_%7B1%7D%2B2%5Calpha%20_%7B2%7D$ ， $%5Cbeta%20_%7B3%7D%3D5%5Calpha%20_%7B1%7D%2B10%5Calpha%20_%7B2%7D-2%5Calpha%20_%7B3%7D$ .