In mathematics, a bilinear operator is a function combining elements of two vector space to yield an element of third vector space that is linear in each of its arguments. Matrix multiplication is an example.
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
such that for any w in W the map
is a linear map from V to X, and for any v in V the map
is a linear map from W to X.
In other words, if we hold t...
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In mathematics, a bilinear operator is a function combining elements of two vector space to yield an element of third vector space that is linear in each of its arguments. Matrix multiplication is an example.
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
such that for any w in W the map
is a linear map from V to X, and for any v in V the map
is a linear map from W to X.
In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.
If V = W and we have B(v,w) = B(w,v) for all v,w in V, then we say that B is symmetric.
The case where X is F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).
The definition works without any changes if instead of vector spaces we use modules over a commutative ring R. It also can be easily generalized to n-ary...
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