## Related questions with answers

Question

Use the axioms of a vector space to prove that

$(a+b)(u+v)=au+av+bu+bv$

for all scalars $a$ and $b$ and all vectors $u$ and $v$ in a vector space.

Solution

VerifiedLet $\bold{u}$ and $\bold{v}$ be a vector in $V$ and $a$ and $b$ scalars. By axiom 8:

$(a+b)(\bold{u}+\bold{v})=a(\bold{u}+\bold{v})+b(\bold{u}+\bold{v})$

By axiom 7:

$(a+b)(\bold{u}+\bold{v})=a\bold{u}+a\bold{v}+b\bold{u}+b\bold{v}$

$\square$

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