Let \(T: V \to W\) be an injective linear transformation and suppose \(v_1, \dots, v_n\) are linearly independent vectors in \(V\). Prove that the vectors \(T(v_1), \dots, T(v_n)\) are linearly independent in \(W\).
Solution
Suppose \[ a_1T(v_1)+\cdots+a_nT(v_n)=0. \] By linearity, \[ T(a_1v_1+\cdots+a_nv_n)=0. \] Since \(T\) is injective, its kernel is \(\{0\}\), so \[ a_1v_1+\cdots+a_nv_n=0. \] The vectors \(v_1,\ldots,v_n\) are linearly independent, hence every \(a_i=0\). Therefore \(T(v_1),\ldots,T(v_n)\) are linearly independent.