Consider a linear transformation T from Rn to Rp and some li

Consider a linear transformation T from R^n to R^p and some linearly dependent vectors v_1dot, v_2dot,...,v_mdot in R^n. Are the vectors T(v_1dot), T(v_2dot),...,T(v_mdot) necessarily linearly dependent? How can you tell?

Solution

Suppose that T:RmRn is a linear transformation and that v₁ ,v₂,…,v_m are linearly dependent vectors in R^m . Then there exist scalars c₁ , c₂ , …,c_m so that c₁ v₁ + c₂ v₂ +… + c_m v_m = 0 where, at least one c_i 0.

Then we get ::

c₁ T(v₁ )+ c₂ T(v₂ )++c_m T(v_m ) = T(c₁v₁ + c₂v₂++c_mv_m) = T (0) = 0

Thus the vectors in Rⁿ are linearly dependent.