Linear transformations and linear independance

Hi,

I was curious as to whether this statement was true or not, If it's true then can anyone show me a hint for the proof of it.

"If we have a linear transformation f:V -> V and if v(1),v(2),...,v(n) are linearly independant vectors in V, then does that imply that f(v(1)),f(v(2)),...f(v(n)) are also linearly independent in V. We are also told that f is injective, so f(v(k)) cannot equal 0, if v(k) is not equal to 0, since we know that with linear transformations f(0)=0."

v(i) means: v subscript i, for all i in [1,2,...,n],

Re: Linear transformations and linear independance

Suppose $\displaystyle a_1f(v_1)+\cdots+a_nf(v_n)=0$. As $\displaystyle f$ is a linear transformation, $\displaystyle a_1f(v_1)+\cdots+a_nf(v_n)=f(a_1v_1+\cdots+a_nv_n)$. Therefore $\displaystyle f(a_1v_1+\cdots+a_nv_n)=0=f(0)$. As $\displaystyle f$ is injective, this implies $\displaystyle a_1v_1+\cdots+a_nv_n=0$, and the linear independence of $\displaystyle v_1,\ldots,v_n$ implies $\displaystyle a_1=\cdots=a_n=0$

Re: Linear transformations and linear independance

Of course, without the condition that the linear transformation is injective, this would be no longer true.