B Show that x v is the only solution to the equation x x

B. Show that x = v is the only solution to the equation x + x = 2v in a vector space V. Cite all axioms used.

8) x+x = 2v

(-x)+x+x = (-x)+2v

Or x = (-x)+2v

Again add -x to both sides

(-x)+x = (-x)+(-x)+2v

Or 0 = (-x)+(-x)+2v (By inverse axiom)

But 0 = (-2v)+2v (By inverse axiom)

By right cancellation

(-x)+(-x) =(-2v)

Cancel -2

x=v is the only solution.

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9) To prove that -0 =0

Since 0 is also a member of vector space V, inverse of 0 is -0 such that

0+(-0) = (-0)+0=0

By left cancellation and right cancellation we have (-0) = 0

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If possible let -v and -v\' be the inverses for v.

Then v-v =0 and v-v\' =0

v-v+0 = v-v (additive axiom)

i.e. v-v +v-v\' =0 (substituting v-v\' for 0)

v-v\' =0

Or -v\' =-v

Or inverse is unique.