(2)不是线性变换.这是因为T(3(1,0,0))=(9,0,0)=(3,0,0)=3T(1,0,0).
(3)是线性变换.因为
∀α,β∈P
3
,可设
α=[e
1
,e
2
,e
3
]X,β=[e
1
,e
2
,e
3
]Y,
则T (kα+lβ)=T[e
1
,e
2
,e
3
](kX+lY)=kT[e
1
,e
2
,e
3
]X+lT[e
1
,e
2
,e
3
]Y
=kTα+lTβ
(4)若f(x
0
)=0,则T是线性变换,否则,T不是线性变换.
这是因为,若f(x
0
)=0,则T是零变换,自然是线性变换;若f(x
0
)=0,则有
T(2f(x))=f(x
0
)=2f(x
0
)=2T(