## 双线性

### 双线性映射

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.

• 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.

• 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
• $$B\left(v_1+v_2, w\right)=B\left(v_1, w\right)+B\left(v_2, w\right)$$
• $$B\left(v, w_1+w_2\right)=B\left(v, w_1\right)+B\left(v, w_2\right)$$
• 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。