Consider a linear transformation T from R^n to R^p and some linearly dependent vectors v1, v2, ..., vm in R^n. Are the vectors T(v1), T(v2), ..., T(vm) necessarily linearly dependent? How can you tell?
Consider a linear transformation T from R^n to R^p and some linearly dependent vectors v1, v2, ..., vm in R^n. Are the vectors T(v1), T(v2), ..., T(vm) necessarily linearly dependent? How can you tell?
Yes. Let $\displaystyle v_m= a_1v_1 + ...+a_{m-1}v_{m-1}$ then $\displaystyle T(v_m)=a_1T(v_1)+...+a_{m-1}T(v_{m-1})$ which shows $\displaystyle T(v_i)$ are linerly dependent