Concept# Bilinear map

Summary

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
B : V \times W \to X
such that for all w \in W, the map B_w
v \mapsto B(v, w)
is a linear map from V to X, and for all v \in V, the map B_v
w \mapsto B(v, w)
is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
Such a map B satisfies th

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