## Vector Spaces!

Linear spaces and subspaces, linear transformations, bases, etc.
andreacmt
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Joined: Sun Nov 14, 2010 5:20 am
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### Vector Spaces!

Hi guys. You know that a vector space V is a set with two operations + and * that satisfy the following properties. However last statement seems to be too obvious (1*u = u) so: Is there a space which satisfies with the first 9 statements and it doesn't satifies the last one (1 * u = u). Thanks or your help!

1)f u and v are elements of V, then u + v is and element of V (closure under +)
2) u + v = v + u
3) u + (v + w) = (u + v) + w
4) There is an element 0 in V such that
u + 0 = 0 + u = u
5) For every u in V there is an element -u with
u + (-u) = 0
6) If u is in V and c is a real number then c*u is in V (closure under *)
7) c * (u + v) = c * u + c * v
8) (c + d) * u = c * u + d * u
9) c * (d * u) = (cd) * u
10) 1 * u = u

stapel_eliz
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### Re: Vector Spaces!

I can't think of anything. To be honest, when I was taking matrix algebra and was testing vector spaces, my first tests would be the zero properties and the 1 properties, because those were where things would "pass" or "fail".

Martingale
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### Re: Vector Spaces!

without 10, how would you prove

$(-1)\cdot u=-u$