Question

Let T:V-W be an injective linear transformation. Prove that if {v1,v2,...,vd} are linearly independent in V then {T(v1),T(v2),..,T(vd)} are linearly independent in W.

Answer 1

Suppose \[\left\{ T(v_1),T(v_2),...,T(v_d) \right\}\]is linearly dependent.
Then there exist scalars a1,a2,...,ad such that
\[a_1T(v_1) + a_2T(v_2) + ... + a_dT(v_d) = 0\]Now, since T is a linear transformation, it satisfies
\[T(av + bw) = aT(v) + bT(w)\]Applying this to the first equation, and noting that T(0) = 0 gives
\[T(a_1v_1 + a_2v_2 + ... + a_dv_d) = 0 = T(0)\]Because T is injective, this equation implies that
\[a_1v_1 + a_2v_2 + ... + a_dv_d = 0\]Which implies that \[\left\{ v_1,v_2,...,v_d \right\}\] are linearly dependent, contradicting the assumption that they are independent.
Hence \[\left\{ T(v_1),T(v_2),...,T(v_d) \right\}\] are also linearly independent.