Question

高数，求解答，计算步骤。第三题中（1）中的两道题。谢谢！

Answer 1

(1)f(x1,x2,x3) = (-2x1+x2+x3)^2+(x1-x2+x3)^2+(x1+x2-2x3)^2
令 C =
-2 1 1
1 -1 1
1 1 -2
因为 |C|=3≠0
所以 C 可逆.
令 Y = CX, 即 X=C^-1Y, 则
f = y1^2+y2^2+y3^2.

(2)f(x1,x2,...,x2n) = x1x2n+x2x2n-1+...+xnxn+1
令 y1 = x1+x2n
y2n = x1-x2n
y2 = x2+x2n-1
y2n-1 = x2-x2n-1
...
yn = xn+xn+1
yn+1 = xn-xn+1
则 f = y1^2+y2^2+...+yn^2 - yn+1^2 - ...-y2n ^2
C =
1 0 ... 0 1
0 1 ... 1 0
... ...
0 1 ... -1 0
1 0 ... 0 -1
Y = CX