Let T Rn Rn be a linear map and a1an is a linearly independ

Let T : Rn Rn be a linear map and {a1,...,an} is a linearly independent set in Rn. Show that if T(ai)=0 for all (1in) then T(v)=0 for all vRn.

Solution

given that vRn.

Rn contains elements {a1,...,an}

so that means v must be equal to ai for some (1in)

then v=ai for some (1in)

then T(v)=T(ai)

T(v)=0    ( given T(ai)=0 for all (1in) )

Hence it is proved that

T(v)=0 for all vRn.