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Recently I come into symplectic vector space and its properties in my linear algebra class.

However, this interesting thing is so different from the usual inner-product spaces I've met before, and I cannot have an intuitive impression of it.

I believe there must be many interesting examples about symplectic spaces, and they may help me understand better~

Coming to the answers: of course the more the better~ thanks~

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    Do you know about the natural symplectic structure on $V \oplus V^{\ast}$ for $V$ a vector space? Up to isomorphism all examples look like this...2012-05-24
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    Not exactly, could you explain it? Thanks~2012-05-24
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    An even more basic example would be $k^{2n}$ with the canonical symplectic form given by the block matrix $\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}$.2012-05-24

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