Can Linear Independence be Proven with Given Information?

[zohapmkoftid]

Homework Statement

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Homework Equations

The Attempt at a Solution

I have no idea how to start. To be linearly independent, c₁u₁+c₂u₂+...+c_nu_n = 0 has only trivial solution. But I don't know how can I use the given information to prove that

 

[HallsofIvy]

You are given that $u_i= Av_i$ for all i.

For any linear combination, $a_1u_1+ a_2u_2+ \cdot\cdot\cdot a_nu_n= 0$ we have $a_1Av_1+ a_2Av_2+ \cdot\cdot\cdot a_nAv_n= A(a_1v_1+ a_2v_2+ a_nv_n)= 0$.

If A is invertible, we have $a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= 0$.

On the other hand, if A is NOT invertible, there exist $v_0\ne 0$ such that $Av_0= 0$ (you should show that). Since $\{v_1, v_2, \cdot\cdot\cdot, v_n\}$ are n independent vectors in $R^n$, we can write $v_0= c_1v_1+ c_2v_2+ \cdot\cdot\cdot+ c_nv_n$ for some numbers $c_n$. Apply A to both sides of that equation.