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## Homework Statement

Show that the positive quadrant

[tex]Q = ( (x,y) | x,y > 0 ) \in \mathbb{R}^2[/tex]

is a vector space.

## Homework Equations

Addition is redefined by

[tex](x_1,y_1) + (x_2,y_2) = (x_1 x_2, y_1 y_2)[/tex]

and scalar multiplication by

[tex] c(x,y) = (x^c , y^c)[/tex]

## The Attempt at a Solution

There are two properties I am having trouble with - the additive identity and additive inverse.

Additive identity

[tex](x,y) + (0,0) = (0,0)[/tex]

which violates the definition that v + 0 = v

Additive inverse

[tex](x,y) + (-x,-y) = (-x^2, -y^2) \not \in \mathbb{V}[/tex]

also violates that v + (-v) = 0

For the identity I might just be thinking about the zero element in the wrong way, but I really have no idea how I could have messed up the additive inverse.

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