# Linear dependent vectors

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• Sep 24th 2009, 12:53 PM
noles2188
Linear dependent vectors
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?
• Sep 24th 2009, 04:36 PM
Jose27
Quote:

Originally Posted by noles2188
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 $v_m= a_1v_1 + ...+a_{m-1}v_{m-1}$ then $T(v_m)=a_1T(v_1)+...+a_{m-1}T(v_{m-1})$ which shows $T(v_i)$ are linerly dependent