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## Homework Statement

Let W

_{1}and W

_{2}be subspaces of a vector space V. Prove that [tex]W_1\oplus{}W_2=V \iff[/tex] each vector in V can be uniquely written as x

_{1}+x

_{2}=v, where [tex]x_1\in W_1[/tex] and [tex]x_2\in W_2[/tex]

## Homework Equations

[tex]W_1\oplus{}W_2=V[/tex] means [tex]W_1\cap W_2 =\{0\}[/tex], [tex]W_1 + W_2 =V[/tex] and W

_{1}& W

_{2}are subspaces of V

8 axioms defining vector space

## The Attempt at a Solution

I'm trying to assume that [tex]\exists x'_1,x'_2: x'_1+x'_2=v[/tex] and [tex]x'_1\in W_1, x'_2\in W_2[/tex] and then proving [tex]x'_1=x_1, x'_2=x_2[/tex], but I'm unsure of where to go from that step.

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