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I would like to seek help in proving that the vector space postulate 1.X = X cannot be derived from the other postulates, e.g. X + 0 = X, X + (Y + Z) = (X + Y) + Z.

The only hint I am given is to construct the "pseudo-scalar product"

c # X = the projection of c.X on a fixed line.

This problem is from Birkhoff and Maclane's "A Survey of Modern Algebra".

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# Prove vector space postulate 1.X = X is independent of others

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