Question
Let $v_1,\cdots,v_n$ be vectors in a vector space $V$ and let $T:V→W$ be a linear transformation.
if $T(v_1),\cdots,T(v_n)$ is linearly independent in $W$, show that $v_1,\cdots,v_n$ is linearly independent in $V$.
Here's what i have so far:
if $T(v_1),\cdots,T(v_n)$ is linearly independent, there exists scalars equal to $0$ such that:
$$c_1T(v_1)+c_2T(v_2)+\cdots +c_nT(v_n)=0\\T(c_1v_1+\cdots+c_nv_n)=0$$
because $T$ is a linear transformation.
Where do I go from here? Do I need to prove that $T$ is injective of can i just state that $v_1,\cdots,v_n$ is linearly independent because I stated earlier that the scalars are equal to $0$?
Best Answer
You want to show $v_1, \ldots, v_n$ are linearly independent. Suppose they are not. Then there are scalars $c_1, \ldots, c_n$ (not all zero) so that $c_1v_1+\ldots +c_nv_n=0$. Then $$ T(c_1v_1+\ldots +c_nv_n)=T(0)=0.$$ So $c_1T(v_1)+\ldots +c_nT(v_n)=0$, which means that $T(v_1), \ldots, T(v_n)$ are not linearly independent. This contradiction means the assumption that the $v_i$s are linearly dependent is false, so they are indeed linearly independent.