Why validate past the first node? In Grokking Bitcoin on page 7, it is noted that:


  Invalid transactions are dropped.  They will not reach further than the first node.


If so, why do nodes after the first node continue to validate the transaction?  If the node is not the first node to receive the transaction, doesn't that mean its valid -- otherwise it would not have reached further than the first node.  

Can't the nodes just receive a broadcast of the tx and save themselves from the redundant verification?
This describes what the first node is supposed to do.  However, there's no guarantee that they will actually do it - they could be running modified software that passes along a transaction even though it is not valid.  The other nodes have no way to ensure that the first node is well behaved, except to redo the check for themselves.

This goes back to the trustless and decentralized nature of Bitcoin.  You don't want to have to rely on other people to avoid being cheated - you want to be able to verify everything independently.