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**Stormey** Let $\displaystyle \varphi :V\rightarrow W$ be a linear transformation, and $\displaystyle v_1, v_2,...v_n\in V$ a set of vectors.

If $\displaystyle v_1, v_2,...v_n$ are linear independent, and $\displaystyle \varphi$ is one-to-one, then $\displaystyle \varphi(v_1), \varphi(v_2),...,\varphi(v_n)$ are also linear independent.

now, since $\displaystyle \varphi$ is a linear transformation, it holds that $\displaystyle \varphi(\vec{0}_V)=\vec{0}_W$, therefore:

$\displaystyle \varphi(\vec{0}_V)=\varphi(\alpha_1 v_1+\alpha_2 v_2+...+\alpha_nv_n)=$

$\displaystyle \alpha_1\varphi(v_1)+\alpha_2\varphi(v_2)+...+ \alpha_n\varphi(v_n)=\vec{0}_W$

and since we know that $\displaystyle \alpha_1=\alpha_2=...=\alpha_n=0$, then $\displaystyle \varphi(v_1), \varphi(v_2),...,\varphi(v_n)$ are linear independent.

**my question here is:** why should I care that $\displaystyle \varphi$ is one-to-one?

any help would be greatly appreciated.