Exercise 28 — Linear Algebra

Prove that for any four vectors $\vec{v}_{1},\ldots,\vec{v}_{4}\in V$ we can associate their sum in any way without changing the result.
$((\vec{v}_{1}+\vec{v}_{2})+\vec{v}_{3})+\vec{v}_{4}=(\vec{v}_{1}+(\vec{v}_{2}+\vec{v}_{3}))+\vec{v}_{4}=(\vec{v}_{1}+\vec{v}_{2})+(\vec{v}_{3}+\vec{v}_{4})\\ =\vec{v}_{1}+((\vec{v}_{2}+\vec{v}_{3})+\vec{v}_{4})=\vec{v}_{1}+(\vec{v}_{2}+(\vec{v}_{3}+\vec{v}_{4}))$
This allows us to write ‘ $\vec{v}_{1}+\vec{v}_{2}+\vec{v}_{3}+\vec{v}_{4}$ ’ without ambiguity.
Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)

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