双线性映射
【双线性映射】(bilinear map)：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.
Let \(V, W\) and \(X\) be three vector spaces over the same base field \(F\). A bilinear map is a function$$ B: V \times W \rightarrow X $$
- 对任意\(w \in W\)，即\(B_w : v \mapsto B(v, w)\)是一个从\(V\)到\(X\)的线性映射；
- 对任意\(v \in W\)，即\(B_v : w \mapsto B(v, w)\)是一个从\(W\)到\(X\)的线性映射。
- 文字表述：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.
例子：
- 矩阵乘法\(M(m, n) \times M(n, p) \rightarrow M(m, p)\)
- 实数域\(\mathbb{R}\)上的线性空间\( V\)的内积，是双线性映射\(V \times V \rightarrow \mathbb{R}\)，注意The product vector space has one dimension.
性质：
- For any \(\lambda \in F, B(\lambda v, w)=B(v, \lambda w)=\lambda B(v, w)\)；
- 映射\(B\) is additive in both components
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- \(B\left(v_1+v_2, w\right)=B\left(v_1, w\right)+B\left(v_2, w\right)\)
- \(B\left(v, w_1+w_2\right)=B\left(v, w_1\right)+B\left(v, w_2\right)\)
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- 如果\(V=W\)，而且对任意\(v, w \in V\)有\(B\left(v, w\right)=B\left(w, v\right)\)，那么我们说映射\(B\)是对称的。进一步地，如果域\(F\)中的元素是我们所谓的scalars，那么映射\(B\) is called a 【双线性形式】(Bilinear form)，下面具体讨论。