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I am having some small trouble with applying the lorentz transformations to calculate lorentz contraction. Here's what I did:

Let O be the rest system and O' be the system moving with velocity v w.r.t O along x axis. Consider a rod lying in the O' system with ends x1' and x2'.

Length of rod in O' system is:

L' = x2' - x1'

measured at the same instant t'.

In the O system,

x2 = ##\gamma##(x2'+vt')

x1 = ##\gamma##(x1'+vt')

So,

L = x2-x1 = ##\gamma##(x2-x1) = ##\gamma##(x2' - x1')

so,

L = ##\gamma##(L')

But... that's not quite right... L should always be smaller than L'...

Where did I go wrong?