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Domanda 5 di 5 (1) Tempo rimanente: 00:54:10 5) Siano dati due vettori non proporzionali \( \mathbf{v}_{1}, \mathbf{v}_{2} \in \mathbb{R}^{3} \). Si consideri un terzo vettore \( \mathbf{v} \in \mathbb{R}^{3} \). Allora: se \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \), allora i tre vettori \( \mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v} \) sono linearmente indipendenti. se \( \mathbf{v} \) è non nullo, allora \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \) se e solo se \( \mathbf{v} \) è il vettore nullo di \( \left.\mathbb{R}_{2}\right) \). se \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \), allora i tre vettori \( \mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v} \) sono linearmente dipendenti.

Ask by Griffiths Christensen. in Italy
Jan 21,2025

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La tercera afirmación es correcta: si \( \mathbf{v} \) está en el espacio generado por \( \mathbf{v}_{1} \) y \( \mathbf{v}_{2} \), entonces los tres vectores \( \mathbf{v}_{1} \), \( \mathbf{v}_{2} \), y \( \mathbf{v} \) son linealmente dependientes.

Solution

Vamos a analizar cada una de las afirmaciones: 1. **Si \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \), entonces los tres vectores \( \mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v} \) son linealmente independientes.** **Falsa.** Si \( \mathbf{v} \) está en el espacio generado por \( \mathbf{v}_{1} \) y \( \mathbf{v}_{2} \), entonces \( \mathbf{v} \) puede expresarse como una combinación lineal de \( \mathbf{v}_{1} \) y \( \mathbf{v}_{2} \). Por lo tanto, los tres vectores son linealmente **dependientes**. 2. **Si \( \mathbf{v} \) es no nulo, entonces \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \) si y solo si \( \mathbf{v} \) es el vector nulo de \( \mathbb{R}^{2} \).** **Falsa.** Hay una inconsistencia en la afirmación: estamos trabajando en \( \mathbb{R}^{3} \), pero se menciona \( \mathbb{R}^{2} \). Además, si \( \mathbf{v} \) es no nulo y está en \( L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \), no puede ser el vector nulo. 3. **Si \( \mathbf{v} \in L\left(\mathbf{v}_{1}, \mathbf{v}_{2}\right) \), entonces los tres vectores \( \mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v} \) son linealmente dependientes.** **Verdadera.** Como \( \mathbf{v} \) se expresa como una combinación lineal de \( \mathbf{v}_{1} \) y \( \mathbf{v}_{2} \), existe una dependencia lineal entre los tres vectores. **Conclusión:** Solo la tercera afirmación es correcta.

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Bonus Knowledge

Immagina di trovarci in un laboratorio di algebra lineare dove i vettori danzano insieme! Quando due vettori \( \mathbf{v}_{1} \) e \( \mathbf{v}_{2} \) non sono proporzionali, significa che non possono essere allineati lungo la stessa direzione. Se aggiungi un terzo vettore \( \mathbf{v} \) che si trova nella combinazione lineare di \( \mathbf{v}_{1} \) e \( \mathbf{v}_{2} \), allora questi tre formano un triangolo ben definito che non può cadere sulla stessa linea: quindi, sono linearmente indipendenti! Ora, parlando di applicazioni nel mondo reale, considera la computer grafica dove i vettori sono essenziali per definire posizioni e direzioni. In questo contesto, se i tuoi vettori rappresentano tre diverse direzioni da cui la luce può arrivare, la loro indipendenza è fondamentale per evitare riflessi e ombre confuse. Solo così puoi ottenere effetti visivi realistici, proprio come in un'opera d'arte!

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