Suppose \(V\) and \(W\) are vector spaces and \(T: V \to W\) is a linear transformation. Suppose \(v_1, \dots, v_k \in V\). Prove that if \(T(v_1), \dots, T(v_k)\) form a linearly independent set in \(W\), then \(v_1, \dots, v_k\) form a linearly independent set in \(V\).
Solution
Suppose \[ a_1v_1+\cdots+a_kv_k=0. \] Applying \(T\) and using linearity gives \[ a_1T(v_1)+\cdots+a_kT(v_k)=0. \] Because \(T(v_1),\ldots,T(v_k)\) are linearly independent, all \(a_i=0\). Hence \(v_1,\ldots,v_k\) are linearly independent.