Let T Rn rightarrow Rm be a linear transformation and V1 V2

Let T: R^n rightarrow R^m be a linear transformation and {V_1, V_2, V_3} be a set of linearly dependent vectors in R^n. Show that [T(V_1), T(V_2), T(V_3)} is a linearly dependent subset of R^m.

Solution

Let a, b and c be non zero real numbers such that

av₁+b v₂+cv₃= 0, as {v_1,v₂,v₃) are linearly dependent vectors.

here a, b and c are not zero.real numbers.

Since, T is linear transformation, so

T(av₁+b v₂+cv₃₎= T(0)

T(av₁)+T(b v₂)+T(cv₃)=0

aT(v₁)+bT( v₂)+cT(v₃)=0, here a, b and c are non zero real numbers.

Thus, [T(v₁),T( v₂),T(v₃)] is a linearly dependent set of R^m.