Exercise I.C.3

Question

Let $V,W$ be vector spaces, and let $\phi:V	o W$ be a linear transformation. Suppose that $v_1,\ldots,v_p\in V$ are such that $\phi(v_1),\ldots,\phi(v_p)$ are linearly independent in $W$ then $v_1,\ldots,v_p$ are linearly independent in $V$.

Toğrul Topal · Accepted Answer

Suppose for the sake of contradiction that $v_1,\ldots,v_p$ are linearly dependent, i.e., there exists a non-trivial collection of scalars $a_1,\ldots,a_p$ such that
    $$a_1v_1 + \ldots + a_pv_p=0$$
    Applying $\phi$ to both sides, we obtain
    $$a_1\phi(v_1) + \ldots + a_p \phi(v_p)=0$$
    a contradiction to the fact that $\phi(v_1),\ldots,\phi(v_p)$ are linearly independent.

Exercise I.C.3

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Exercise I.C.3

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