Multilinear algebra

We now proceed into a new territory. So far, we have been looking at linear structure, called vector spaces, and linear maps between them. When working with these spaces, we are allowed to add vectors and multiply vectors by scalars. But there are times where we would like to multiply vectors together. The inner product

$$ v \cdot w = v_1 w_1 + v_2 w_2 + \cdots + v_n w_n $$

defined on $k^n$ is one example. Another interesting example is the cross product

$$ v \times w = (v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2 w_1) $$

defined on $k^3$. More generally, notions such as “area” or “volume” necessarily involve multiplying vectors together in some way. All multiplications here satisfy the property that the map is linear if all but one component is fixed.