True or False. Justify your answer. Suppose T: R n ---> R m is a lineartransform

Question

True or False. Justify your answer.

Suppose T: Rn ---> Rm is a lineartransformation and that {X1,...,Xk} is alinearly independent set of vectors in Rn. Then{T(X1),......T(Xk)} is a linearly independentset of vectors in Rm.

I thought this was true, but the answer says its False...
It said smth about T = 0 ==> T(Xk) = 0

Thanks.

Explanation / Answer

True or False. Justify your answer. Suppose T: Rn ---> Rm is a lineartransformation and that {X1,...,Xk} is alinearly independent set of vectors in Rn. Then{T(X1),......T(Xk)} is a linearly independentset of vectors in Rm. I thought this was true, but the answer says its False... It said smth about T = 0 ==> T(Xk) = 0 DIMENSION OF RN IS N AND IT REQUIRES N INDEPENDENTVECTORS TO FORM ITS BASIS AND IT CAN NOT HAVE MORE THAN N INDEPENDENT VECTORS. SO IN THE GIVEN SET OF K LINEARLY INDEPENDENT VECTORS{X1,...,Xk} ......K

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