Question

微分几何，求解答？

Answer 1

（1）证明：对∀(x1;y1)，(x2;y2)∈V，∀k∈数域P，有
T(x1;y1)+T(x2;y2)
=(x1-y1; x1+y1; 2x1)+(x2-y2; x2+y2; 2x2)
=(x1+x2-y1-y2; x1+x2+y1+y2; 2x1+2x2)
=T(x1+x2; y1-y2)
T[k*(x1;y1)]

=T(kx1;ky1)
=(kx1-ky1; kx1+ky1; 2kx1)
=k*(x1-y1; x1+y1; 2x1)
=k*T(x1;y1)
所以T为V上的线性变换
（2）令T=(T11,T12; T21,T22; T31,T32)
因为T(x;y)=(x-y;x+y;2x)，所以
①T11*x+T12*y=x-y
②T21*x+T22*y=x+y
③T31*x+T32*y=2x
所以T11=1，T12=-1，T21=1，T22=1，T31=2，T32=0
即T=(1,-1; 1,1; 2,0)