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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.

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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: $B(a,b) = a\cdot b$

$B(x+y,z) = (x+y)\cdot z = x\cdot z + y\cdot z = B(x,z) + B(y,z)$

$B(x,y+z) = x\cdot (y+z) = x\cdot y + x\cdot z = B(x,y) + B(x,z)$.

$B(cx,z) = (cx)\cdot z = c\cdot(xz) = x\cdot(cz) = B(x,cz)$

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As RobertIsrael already mentioned, a bilinear map is just a mapping which is linear in two variables.

If you have a linear map $L$ which maps a vector space $X$ into another space $Y$ then you can write,

$$ L( a \vec{u} + b \vec{v} ) = a L( \vec{u} ) + b L (\vec{v}) $$.

With a bilinear map you are mapping from a cartesian product of vector spaces to some other vector space. So let $B: X \times Y \rightarrow Z$ be a bilinear map. Then we can write,

$$ B(a \vec{u} + b \vec{v},c \vec{s} + d \vec{t}) = ac B(\vec{u},\vec{s}) + ad B(\vec{u},\vec{s}) + bc B( \vec{v}, \vec{s}) + bd B(\vec{v},\vec{t}),$$

A simple example of such a map is the mapping $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}:(x,y)\rightarrow x\cdot y$. Where we consider the real numbers to be a vector space over the real field.

Another example of such a mapping is the inner product (when its restricted to only have real values).

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Do you know what a linear map is? A bilinear map is a map in two variables (each of which could take values in some vector space) that is linear in each separately. That is, $B(x,y)$ is a bilinear map if for each $x$, the map taking $y$ to $B(x,y)$ is linear and for each $y$, the map taking $x$ to $B(x,y)$ is linear.

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