Related to linear map 6 In reference to the previous exercis

Related to linear map

6. In reference to the previous exercise, we say that if V and W are vector spaces, then a function T V W is called additive if T(u v) T(n) T(v) for all u, v E V

Solution

Given that T(u+v) = T(u)+T(v) for all u,v in V.

Case I:

Let n be a natural number.

Hence T(nu) = T(u+u+u+...n times)

= T(u)+T(u)+...+T(u) n times

= n T(u) for n being a natural number

Case II: Let n =0

T(0.u) = T(0) =0

Case III: Let n be a negative integer

Then n = -m

where m is natural number

T(nu) = T(-mu) = T(0-mu)

= T(0) -mT(u)

= -mT(u)

= nT(u)

Hence true for all integers.