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.