### 域代数

(非)结合代数：根据乘法是否满足结合律可分为结合代数和非结合代数。比如三维欧氏空间中的叉乘就不满足结合律，而矩阵的乘法满足结合律。
酉代数：如果乘法有单位元，则称为酉(unital/unitary)。

Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring.

### 具体实例

is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

Linear transformations of V into V are often called linear operators on V

### 多重线性代数

multivector
p-vector
bivector

Linear Map $f: V \rightarrow W$
Multilinear Map f: V^n^ \rightarrow W $n=2$, bilinear; $n=3$, trilinear

### 张量代数

The tensor algebra T(V) is a functor from K-Vect, the category of vector spaces over K, to K-Alg, the category of K-algebras.
T(V) is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the “most general” algebra containing V, in the sense of the corresponding universal property.

It is the quotient of the tensor algebra TV by the relation

### 克利福德(Clifford)代数

1 scalar component
3 vector components
3 bivector components, which correspond to the 3 linearly independent planes in a 3d space
1 trivector or pseudoscalar component, which corresponds to the single, oriented unit volume in 3d space

### 几何代数(Geometric algebra)

covectors

[1]：(双)线性运算简单来说就是保持加法和标量乘法的运算。双线性运算特殊在它是二元运算，即二元线性运算：由于是二元运算，需要考虑左结合和右结合，对前后两个元都要是线性的，即双线性。

或者合在一起 $f(ax+by) = af(x) + bf(y)$

$\qquad\quad~f(x, b_1y_1+b_2y_2) = b_1f(x,y_1) + b_2f(x,y_2)$