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Let $V$ be a vector space, $T : V \to V$ a linear transformation, and $v_1, v_2, v_3$ be nonzero vectors in $V$ such that $T(v_1)=v_1, T(v_2)=v_1 + v_2, T(v_3)=v_2 + v_3.$ Prove that the vectors $v_1, v_2, v_3$ are linearly independent.

I'm confused as to where to start. Does it involve using the properties of a linear transformation?

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A typical method for showing linear independence is to assume dependence and derive a contraction. Suppose the vectors are dependent, so that there are constants $\alpha_i$ not all zero so that $$\alpha_1 v_1 + \alpha_2 v_2 + \alpha_3 v_3 = 0 .$$ Now, apply $T$ and use linearity.

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  • $\begingroup$ So I apply T to each one, then I'm left with a1v1+a2(v2+v3)+a3(v2+v3)=0. I rearranged this to be v1(a1)+v2(a2+a3)+v3(a2+a3)=0. How am I supposed to show that these must equal 0? $\endgroup$ – user417408 Mar 12 '17 at 23:42
  • $\begingroup$ You now have a system of two equations in $a_1, a_2, a_3$. Can you combine the equations to produce simpler one and deduce some information? (By the way, the second term in your first equation should be $\alpha_2 (v_1 + v_2)$.) $\endgroup$ – Travis Willse Mar 12 '17 at 23:46

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