Linear in each slot separately
Let $R$ be a commutative ring and $M$, $N$, $P$ be $R$-modules. A bilinear map $b: M \times N \to P$ is a function that is linear in each variable when the other is held fixed. Concretely, for all $m, m' \in M$, $n, n' \in N$, $r \in R$: $b(m+m', n) = b(m,n) + b(m',n)$, $b(rm, n) = r\,b(m,n)$, and the symmetric pair of conditions in the second slot. If you already know vector spaces, just read ‘module’ as ‘vector space’ on a first pass — nothing in this guide breaks.
A multilinear map is the same idea with $k$ slots: $f: M_1 \times \cdots \times M_k \to P$ is linear in each $M_i$ separately. The dot product, the cross product, matrix multiplication, the determinant as a function of its columns — all are multilinear. The whole subject is, in a sense, the systematic study of these maps.
How much data is a bilinear map?
Over a field, fix bases $(e_i)$ of $M$ and $(f_j)$ of $N$. Bilinearity says a bilinear map is completely determined by its values $b(e_i, f_j)$ on pairs of basis vectors, and you may choose those values freely. So a bilinear form $b: M \times N \to k$ with $\dim M = m$, $\dim N = n$ is exactly a matrix of size $m \times n$ — the array $B_{ij} = b(e_i, f_j)$. This counting, $mn$ free parameters, is the first hint that bilinear maps want to live on a space of dimension $mn$, not $m + n$.
Example: a bilinear form on R^2. Take M = N = R^2 with basis e_1, e_2.
A bilinear form b is fixed by the four numbers
B = [ b(e_1,e_1), b(e_1,e_2); b(e_2,e_1), b(e_2,e_2) ].
Then for x = x_1 e_1 + x_2 e_2 and y = y_1 e_1 + y_2 e_2,
b(x, y) = sum_{i,j} x_i B_{ij} y_j = x^T B y (row vector x^T, column y).
Concrete dot product: B = [1, 0; 0, 1] gives b(x,y) = x_1 y_1 + x_2 y_2.
A skew example: B = [0, 1; -1, 0] gives b(x,y) = x_1 y_2 - x_2 y_1
= det([x_1, y_1; x_2, y_2]).
Note b(x,x) = 0 here: this alternating form will reappear as the wedge product.Two special kinds of bilinear form deserve names now and a track of their own later: a symmetric form has $b(x,y) = b(y,x)$ (matrix $B = B^T$), and a skew or alternating form has $b(x,x) = 0$ (forcing $b(x,y) = -b(y,x)$ away from characteristic $2$). The skew row above is the seed of the entire exterior algebra.
The dream: turn ‘bilinear’ into ‘linear’
Linear algebra is powerful precisely because linear maps factor through matrices, compose cleanly, and have kernels and images. Bilinear maps have none of that machinery directly. So here is the program for the rest of the track: build a single space $M \otimes N$ and a single bilinear map $\otimes: M \times N \to M \otimes N$ so universal that every bilinear map out of $M \times N$ becomes an ordinary linear map out of $M \otimes N$. Then the entire toolbox of linear algebra applies to multilinear problems.