I am trying to prove that, given two linearly independent vectors v_1,v_2, the sum of vectors v_3+v_4 (both being some linear combination of v_1,v_2) is linearly dependent with v_1,v_2. It makes sense to me intuitively since if we have x_1v_1+x_2v_2=v_3 and y_1v_1+y_2v_2=v_4, then v_3+v_4=x_1v_1+x_2v_2+y_1v_1+y_2v_2, but I'm struggling to formulate a proof using the definition of linear dependency. Any suggestions on how I might do so?

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