TITLE: Definition of Bilinear maps.
QUESTION [7 upvotes]: Please I need a layman's definition of the bilinear map.
Wikipedia has a brilliant one, I however don't seem to make anything of it.

REPLY [16 votes]: Loosely speaking, a bilinear map satisfies:
$B(x+y,z) = B(x,z) + B(y,z)$ (additive in the first "coordinate"),
$B(x,y+z) = B(x,y) + B(x,z)$ (additive in the second "coordinate"),
$B(cx,y) = cB(x,y) = B(x,cy)$ (preserves scaling in each "coordinate").
Think about $B$ as multiplication of real numbers for example: $B(a,b) = a\cdot b$
$B(x+y,z) = (x+y)\cdot z = x\cdot z + y\cdot z = B(x,z) + B(y,z)$
$B(x,y+z) = x\cdot (y+z) = x\cdot y + x\cdot z = B(x,y) + B(x,z)$.
$B(cx,z) = (cx)\cdot z = c\cdot(xz) = x\cdot(cz) = B(x,cz)$