# Bilinear Map Let $V$, $W$ and $X$ be three [vector spaces](Abstract%20Vector%20Spaces.md) over the same base [Field](Field.md) $F$. A **bilinear map** is a function: $B : V \times W \rightarrow X$ Such that for all $w \in W$, the map: $B_w: v \mapsto B(v, w)$ is a [linear map](Linear%20Transformations.md) 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. ### Example 1 - Matrix Multiplication Matrix multiplication is a bilinear map: $M(m, n) \times M(n, p) \rightarrow M(m, p)$ Remember, [the set of matrices of fixed size form a vector space](Abstract%20Vector%20Spaces.md). ### Example 2 - Inner Product If a vector space $V$ over the real numbers $\mathbb{R}$ carries an [Inner Product](Inner%20Product.md), then the inner product is a bilinear map $V \times V \rightarrow \mathbb{R}$. --- Date: 20211214 Links to: Tags: References: * [Wikipedia](https://en.wikipedia.org/wiki/Bilinear_map)