1009: Dual Basis for Covectors (Dual) Space of Basis for Finite-Dimensional Vectors Space

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definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of covectors (dual) space of vectors space.

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Target Context

• The reader will have a definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V$: $\in \{\text{ the finite-dimensional }F\text{ vectors spaces }\}$
$V^{\ast }$: $=L(V:F)$
$J$: $\in \{\text{ the finite index sets }\}$
$B$: $\in \{\text{ the bases for }V\}$, $=\{b_j|j\in J\}$
$\ast B^{\ast }$: $\in \{\text{ the bases for }{V}^{\ast }\}$, $=\{b^j|j\in J\}$
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Conditions:
$\mathrm{\forall }j\in J(\mathrm{\forall }k\in J(b^j(b_k)={\delta }_k^j))$
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2: Note

Let us see that $B^{\ast }$ is indeed a basis for $V^{\ast }$.

Let $w\in V^{\ast }$ be any. For each $j\in J$, $w_j:=w(b_j)\in F$. For each $v\in V$, $v=v^j{b}_j$, and $w(v)=w(v^j{b}_j)=v^{j}w(b_j)=v^j{w}_j$. $w_j{b}^j(v)=w_j{b}^j(v^k{b}_k)=w_j{v}^k{b}^j(b_k)=w_j{v}^k{\delta }_k^j=w_j{v}^j$. So, $w=w_j{b}^j$, which means that $B^{\ast }$ spans $V^{\ast }$.

Let $c_j{b}^j=0$. $c_j{b}^j(b_k)=0(b_k)=0$, but $c_j{b}^j(b_k)=c_j{\delta }_k^j=c_k$, so, $c_k=0$ for each $k\in J$, which means that $B^{\ast }$ is linearly independent.

$V^{\ast }$ inevitably has the dimension of $V$.

For each $w\in V^{\ast }$, the components of $w$ with respect to $B^{\ast }$ are $w(b_1),...,w(b_d)$.

It is crucial that $V$ is finite-dimensional: otherwise, $w_j{b}^j$ may not make sense, because some infinitely many $w_j$ s may be nonzero.

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References

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