Geometry in mechanics: Linear space

Mar 2, 2024

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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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