Let T be a linear transformation Show that if vector v1 vect

Let T be a linear transformation. Show that if vector v_1, vector v_2, and vector v_3, are linearly dependent then so are T(vector v_1), T(vector v_2), and T(vector v_3).

Solution

v1,v2,v3 are linearly dependent so there exist, a,b,c not all 0 so that

av1+bv2+cv3=0

T(av1+bv2+cv3)=aT(v1)+bT(v2)+cT(v3)=0

Hence, T(v1),T(v2),T(v3) are linearly independent