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Solution to Linear Algebra Done Wrong Exercise 1.1.7

Prove that $0{\bf v}={\bf 0}$ for any vector ${\bf v}\in V$.

Solution: We have
0{\bf v}=&\ 0{\bf v}+{\bf 0}\\ =&\ 0{\bf v}+({\bf v}+(-{\bf v}))\\
=&\ (0{\bf v}+{\bf v})+(-{\bf v})\\ =&\ (0{\bf v}+1{\bf v})+(-{\bf v})\\
=&\ (0+1){\bf v}+(-{\bf v})\\=&\ 1{\bf v}+(-{\bf v})\\ =&\ {\bf v}+(-{\bf v})={\bf 0}.

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