Espai Dual

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Recursos externs

Since I haven't posted for a while, I decided to break up my rants about homology with some posts on linear (and multilinear) algebra. In this post, we will (as usual) deal only with finite dimensional vector spaces.

https://algebrology.github.io/dual-spaces/

Covector: The Long-Lost Twin of Vector?

First of all, let us review how the vector components change with different basis vectors. Consider the following diagram with unprimed basis vectors (blue) and primed basis vectors (green). Suppos…

https://almosttensorial.wordpress.com/covector-the-long-lost-twin-of-vector/

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp...

https://math.stackexchange.com/questions/1286100/how-do-i-find-a-dual-basis-given-the-following-basis

Understanding dual spaces

Time for a short preparatory post about vector spaces. This is a follow up of my differential form post…

https://medium.com/@luca.ambrogioni/understanding-dual-spaces-4b39d000a7ed

On the Visualization of the Dual Space

This post talks about visualization of the dual space $V^*$. To read it, you need to know what a vector space is. If you don’t, this post is not for you.

https://web.tecnico.ulisboa.pt/~ist189623/wordpress/2021/08/08/on-the-visualization-of-the-dual-space/

Espai Vectorial - Repàs

Informació al respecte

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Estructures algebraiques així pim pam

Espai Vectorial

Producte Intern, norma, mètrica… i altres criatures fantàstiques

Definició d’Espai Dual

Visualització de covectors

Quan hi ha definit un producte intern hi ha un isomorfisme natural entre i , això vol dir que a cada vector se li pot assignar un covector i viceversa.

Per exemple, el vector , té un covector assignat tal que . En altres paraules, si coneixem un vector (i per tant el seu mòdul), coneixem el seu corresponent covector.

En l’espai vectorial el covector serà un vector amb les mateixes components (pel cas euclidià), i en l’espai vectorial el covector serà un conjunt de línies paral·leles. Aquestes seran tals que passen per i toquen tan l’eix com l’eix , en el punt . (El número en el nostre exemple visual).