## Linear spacce

In short, a linear space (also called vector space) is closed under ‘addtion’ and ‘scalar multiplication’ operation.

**space**

To define a linear space (also called vector space), we will firstly need a number field `F`

and a non-empty set `\bold{V}`

.

The number field can be integer, real number or complex number, etc.

The set could be anything actually. Such as a ordered array, matrix, tensor, function…

**Addition and scalar multiplication**

You can call whatever operations you defined as ‘addtion’/’scalar multiplication’ unless these eight axioms are satisfied.

Suppose `\alpha,\beta\in F`

and `\boldsymbol{a,b,c}\in\bold{V}`

```
\begin{align*}
\text{comutativity of addition}&:\bold{a}+\bold{b}=\bold{b}+\bold{a}\\
\text{associativity of addition}&:\bold{a}+(\bold{b}+\bold{c})=(\bold{a}+\bold{b})+\bold{c}\\
\text{existence of zero element}&:\exists\bold{0}\in\bold{V}, \bold{a}+\bold{0}=\bold{a}\\
\text{existence of negative element}&:\exists-\bold{a}\in\bold{V}, \bold{a}+(-\bold{a})=\bold{0}\\
\text{distributivity of scalar}&:(\alpha+\beta)\bold{a}=\alpha\bold{a}+\beta\bold{a}\\
\text{distributivity of vector}&:\alpha(\bold{a}+\bold{b})=\alpha\bold{a}+\alpha\bold{b}\\
\text{comutativity of scalar multiplication}&:\alpha\beta\bold{a}=\beta\alpha\bold{a}\\
\text{existence of identity number}&:\exists1\in F,1\cdot\bold{a}=\bold{a}
\end{align*}
```

**Close under addition and multiplication**

Linear space is closed under addition and scalar multiplication, which means new elements generated by operation always lay in the set `\bold{V}`

Let’s introduce some important concepts of linear space.

## Linear independence

Suppose we have `n`

vectors `\bold{a}_1,\bold{a}_2,\cdots,\bold{a}_n`

. Linear indenpendence means any vector can not be expressed by other vectors. Or we can see the following equation

`x_1\bold{a}_1+x_2\bold{a}_2+\cdots+x_n\bold{a}_n=\bold{0}`

Only has trivial solution.

In other words, non trivial solution means these vectors are linear dependent.

For one linear space, the number of vectors in a maximal linearly independent group is the dimension of space.

## Base and coordinate

Given a linear space, once we find a maximal linearly independent group (`\bold{e}_{1,\dots,n}`

), we can use this group of vectors to express any other vectors as

`\bold{x}=x^i\bold{e}_i`

where `\bold{e}_i`

is called base vector, `x^i`

is called coordinate. Here Einstein summation convention is applied: same sub-/sup- indices means summation.

Given base vectors, every vector has unique coordinates.

*Proof*: We assume vector `\bold{x}`

has two different coordinates like `\bold{x}=x^i\bold{e}_i=y^i\bold{e}_i`

. We subtract two of them

`\bold{0}=(x^i-y^i)\bold{e}_i`

Because `\bold{e}_i`

are linear independent, above equation only has trivial solution, which is `x^i-y^i=0`

## Transformation of base and coordinate

Base vectors of a linear space is nonunique. Suppose we have old base vector `\bold{e}_i`

and new base vectors `\bar{\bold{e}}_{i'}`

.

We can express new base by old ones:

`\bar{\bold{e}}_{i'}=\alpha^i_{i'}\bold{e}_i`

We use two base to express same vector and therefore get coordinate transformation,

`\bold{x}=x^{i'}\bar{\bold{e}}_{i'}=x^{i'}\alpha^i_{i'}\bold{e}_i=x^i\bold{e}_i`

i.e.

`x^i=\alpha^i_{i'}x^{i'}`

It is very interesting that we express new base by old base with `\alpha^i_{i'}`

. But with same `\alpha^i_{i'}`

we can express old coordinate by new coord. This is called inverse relation.

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