diff --git a/src/Data/AdditiveGroup.hs b/src/Data/AdditiveGroup.hs
--- a/src/Data/AdditiveGroup.hs
+++ b/src/Data/AdditiveGroup.hs
@@ -102,6 +102,36 @@
   Just a' ^+^ Just b' = Just (a' ^+^ b')
   negateV = fmap negateV
 
+{-
+
+Alexey Khudyakov wrote:
+
+  I looked through vector-space package and found lawless instance. Namely Maybe's AdditiveGroup instance
+
+  It's group so following relation is expected to hold. Otherwise it's not a group.
+  > x ^+^ negateV x == zeroV
+
+  Here is counterexample:
+
+  > let x = Just 2 in x ^+^ negateV x == zeroV
+  False
+
+  I think it's not possible to sensibly define group instance for
+  Maybe a at all.
+
+
+I see that the problem here is in distinguishing 'Just zeroV' from
+Nothing. I could fix the Just + Just line to use Nothing instead of Just
+zeroV when a' ^+^ b' == zeroV, although doing so would require Eq a and
+hence lose some generality. Even so, the abstraction leak would probably
+show up elsewhere.
+
+Hm.
+
+-}
+
+
+
 
 -- Memo tries
 instance (HasTrie u, AdditiveGroup v) => AdditiveGroup (u :->: v) where
diff --git a/src/Data/LinearMap.hs b/src/Data/LinearMap.hs
--- a/src/Data/LinearMap.hs
+++ b/src/Data/LinearMap.hs
@@ -1,5 +1,5 @@
 {-# LANGUAGE TypeOperators, FlexibleContexts, TypeFamilies, GeneralizedNewtypeDeriving, StandaloneDeriving #-}
--- {-# OPTIONS_GHC -Wall -fno-warn-orphans #-}
+{-# OPTIONS_GHC -Wall -fno-warn-orphans #-}
 ----------------------------------------------------------------------
 -- |
 -- Module      :  Data.LinearMap
@@ -17,6 +17,7 @@
    , inLMap, inLMap2, inLMap3
    , liftMS, liftMS2, liftMS3
    , liftL, liftL2, liftL3
+   , firstL
    )
   where
 
@@ -46,6 +47,28 @@
 newtype u :-* v = LMap { unLMap :: LMap' u v }
 
 deriving instance (HasTrie (Basis u), AdditiveGroup v) => AdditiveGroup (u :-* v)
+
+instance (HasTrie (Basis u), VectorSpace v) =>
+         VectorSpace (u :-* v) where
+  type Scalar (u :-* v) = Scalar v
+  (*^) s = (inLMap.liftMS.fmap) (s *^)
+
+firstL :: ( HasBasis u, HasBasis u', HasBasis v
+          , HasTrie (Basis u), HasTrie (Basis v) 
+          , Scalar u ~ Scalar v, Scalar u ~ Scalar u'
+          ) =>
+          (u :-* u') -> ((u,v) :-* (u',v))
+firstL = linear.first.lapply
+
+-- TODO: more efficient firstL
+
+-- liftMS :: (AdditiveGroup a) => (a -> b) -> (MSum a -> MSum b)
+
+-- (s *^) :: v -> v
+-- fmap (s *^) :: (Basis u :->: v) -> (Basis u :->: v)
+-- (liftMS.fmap) (s *^) :: LMap' u v -> LMap' u v
+-- (inLMap.liftMS.fmap) (s *^) :: (u :-* v) -> (u :-* v)
+
 
 -- Before version 0.7, u :-* v was a type synonym, resulting in a subtle
 -- ambiguity: u:-*v == u':-*v' does not imply that u==u', since Basis
diff --git a/src/Data/VectorSpace.hs b/src/Data/VectorSpace.hs
--- a/src/Data/VectorSpace.hs
+++ b/src/Data/VectorSpace.hs
@@ -27,7 +27,7 @@
   ( module Data.AdditiveGroup
   , VectorSpace(..), (^/), (^*)
   , InnerSpace(..)
-  , lerp, magnitudeSq, magnitude, normalized
+  , lerp, magnitudeSq, magnitude, normalized, project
   ) where
 
 import Control.Applicative (liftA2)
@@ -81,6 +81,11 @@
 -- given the zero vector, then return it.
 normalized :: (InnerSpace v, s ~ Scalar v, Floating s) =>  v -> v
 normalized v = v ^/ magnitude v
+
+-- | @project u v@ computes the projection of @v@ onto @u@.
+project :: (InnerSpace v, s ~ Scalar v, Floating s) => v -> v -> v
+project u v = (v <.> u') *^ u'
+  where u' = normalized u
 
 instance VectorSpace Double where
   type Scalar Double = Double
diff --git a/vector-space.cabal b/vector-space.cabal
--- a/vector-space.cabal
+++ b/vector-space.cabal
@@ -1,5 +1,5 @@
 Name:                vector-space
-Version:             0.7.3
+Version:             0.7.6
 Cabal-Version:       >= 1.2
 Synopsis:            Vector & affine spaces, linear maps, and derivatives (requires ghc 6.9 or better)
 Category:            math
@@ -20,7 +20,7 @@
 Maintainer:          conal@conal.net
 Homepage:            http://haskell.org/haskellwiki/vector-space
 Package-Url:         http://code.haskell.org/vector-space
-Copyright:           (c) 2008-2010 by Conal Elliott
+Copyright:           (c) 2008-2011 by Conal Elliott
 License:             BSD3
 License-File:        COPYING
 Stability:           experimental