Linear Transformations regarding Linear Dependence

Question

Suppose $V$ and $W$ are vector spaces, and suppose $T : V \to W$ is a linear transformation. Let $S = \{v_1,v_2,v_3\}$ be a set of linearly dependent vectors in $V$ .

Prove that $S' = \{T(v_1), T(v_2), T(v_3)\}$ is a linearly dependent set of vectors in $W$.

Answer 1

By definition $\exists a,b,c\neq(0,0,0)$ such that

$$av_1+bv_2+cv_3=0$$

then

$$T(av_1+bv_2+cv_3)=aT(v_1)+bT(v_2)+cT(v_3)=0$$

thus also $T(v_1),T(v_2), T(v_3)$ are linearly dependent.

Answer 2

The vectors are linearly dependents iff there exist $(\lambda_1,\lambda_2,\lambda_3) \neq (0,0,0)$ such that: $$ \lambda_1 v_1 +\lambda_2 v_2 +\lambda_3 v_3=0$$ Using the linearity of $T$ we obtain: $$\lambda_1 T(v_1) +\lambda_2 T(v_2) +\lambda_3 T(v_3)=T( \lambda_1 v_1 +\lambda_2 v_2 +\lambda_3 v_3)=0$$ so $(T(v_1),T(v_2),T(v_3))$ are linearly dependents.