The vector space with the operation (+) over a field of real number is an abelian group with a scalar multiplication. So, suppose , and , they satisfy the following conditions:

Abelian group:

- Closure: ,
- Associativity: ,
- Identity: such that and ,
- Invertibility such that and ,
- Commutativity: .

Scalar multiplication:

- , such that ,
- , such that and ,
- .

This is not a tutorial, please refer to the books for tensor algebra.

**Linearly independent:**

For a set of non-zero vectors , , such that,

if and only if .

Such set of the vectors is called linearly independent.

**Vector space dimension:**

Let be a subset of a vector space and , the vector,

is the linear combination of the vectors .

With the definition of the linear combination, the vector space dimension can be defined: A set of linearly independent vectors is a basis of a vector space if every vector in is the linear combination of the elements in .

: All the basis of a finite-dimensional vector space contains the same number of vectors.Theorem

: For aTheoremn-dimension vector space , every set contains n linearly independent vectors is a basis of , every set contains more thannvectors is linearly dependent.

**Summation convention:**

Let be the basis of the vector space , any vector in such space can be represented with the linear combination of the basis,

The expression is shorten without the summation symbol. This is called the Einstein summation. As shown on the mathworld, there are three rules of this notational convention:

- Repeated indices are implicitly summed over,
- Each index can appear at most twice in any term,
- Each term must contain identical non-repeated indices.

*Theorem*: The representation of the vector in the vector space with respect to a given basis is unique.

**Euclidean space:**

Suppose , and , then the vector space with inner product of the vectors satisfying the following conditions is called Euclidean space :

- Commutativity: ,
- Distributivity: ,
- Associativity with a scalar: ,
- , if and only if .

Since the inner product of the vectors in is non-negative, the Euclidean distance or length (Norm) is defined by,

**Orthonormal basis:**

- Orthogonal: ,
- Normal: .

Let be a basis of , if satisfy the preceding conditions, then the set is called orthonormal basis, i.e.

,

The symbol is called Kronecker delta,

Every set of linearly independent vectors in Euclidean space can be orthogonalized and normalized with the Gram-Schmidt procedure.

**Dual basis:**

The dual basis of vectors is a biorthogonal system of the original basis of vectors in vector space , the detail information can be found in wikipedia. It is a little bit abstract, the following figure shows the original basis and the dual basis in 2D Euclidean space .

In this figure, . Now, extend the 2D Euclidean space to *n*D Euclidean space , the basis is the dual basis to the basis , if that,

.

: There exists a unique dual basis of each basis in an Euclidean space .Theorem

: Let and be the two dual bases of the original basis , recall the properties of the dual basis,Proofand

Expand we get,

Then,

Now, let be the basis of , then,

Expend the with respect to the , then,

Thus,

we get,

Then, the matrices and (i.e. ) are **inverse** of each other.

**Dual basis Properties:**

For orthonormal basis of , it is a self dual vector space,

and

For non-orthonormal basis of ,

and

The inner product of vector and ,

**Levi-Civita symbol:**

Levi-Civita symbol is the permutation symbol,

- Even permutation: ,
- Odd permutation: ,
- Otherwise: more than two symbols are the same e.g. .

Let be the orthonormal vectors in and [] be the symbol of mixed product of three vectors,

Denote *g* as the result for mixed product of basis vectors in ,

The last term is the determinant of the matrix i.e. ,

Since we get,

Thus,

.

**Levi-Civita symbol properties:**

**Mixed product (triple product):**

With the permutation symbol, the mixed product can be expressed by,

then multiplying both sides of this relation by the vector ,

.

Since ,

Then we get,

Thus,

.

Then,

For 3D Euclidean space , , the cross product is,

hi it not better first you define a vector space on a filde with some example then go to define tensor

Hi, hardanian, thanks for your advice. I will make it better~

thanks so much tomorrow i wil come to study it again