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_{1}and W

_{2}in symbols:

V = W

_{1}(+) W

_{2}if:

V = W

_{1}+ W

_{2}and

W

_{1}[itex]\cap[/itex] W

_{2}= {0}

Show that the following statements are equivalent:

1. V = W

_{1}(+) W

_{2}

2. Every vector v[itex]\in[/itex]V can be written uniquely as w

_{1}+ w

_{2}where w

_{1}[itex]\in[/itex]W

_{1}and w

_{2}[itex]\in[/itex]W

_{2}

3. V = W

_{1}+ W

_{2}and for vectors w

_{1}[itex]\in[/itex]W

_{1}and w

_{2}[itex]\in[/itex]W

_{2}if w

_{1}+w

_{2}= 0 then w

_{1}= w

_{2}

4. If [itex]\alpha[/itex]

_{1}is a basis for W

_{1}and [itex]\alpha[/itex]

_{2}is a basis for W

_{2}then

[itex]\alpha[/itex] = [itex]\alpha[/itex]

_{1}[itex]\cup[/itex] [itex]\alpha[/itex]

_{2}

Attempt:

I'm thrown off here because I'm given the definition of a direct sum. So how can I even show that all these statements are equivalent? Would I start by assuming one of the statments is true? For example: Let's say statement 1 is true. that means......well it means exactly what statement 2 is. Maybe take a vector in W

_{1}and one in W

_{2}sum them together and create a vector in V? But then to do that I would have to define an exact vector space.......