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4. Which tool(s) should a researcher use to locate information within a general reference work? the guide words at the top of the page the index the table of contents the chapter headings

Ask by Franklin Matthews. in the United States
Jan 21,2025

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Use the table of contents, index, guide words, and chapter headings to find information in a general reference work.

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Bonus Knowledge

To effectively navigate a general reference work, researchers should definitely start with the table of contents. It's like a treasure map for the entire book, outlining the main topics and the pages where they can be found. This helps you get a sense of what's inside without flipping through every page like a lost tourist! Additionally, utilizing the index at the back of the book is a game-changer. It’s like having a personal assistant to point you directly to specific topics or keywords, saving you precious time. Just imagine diving straight to your desired nugget of information without all the unnecessary fluff—now that's efficiency!

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Numéro d'étudiant : 22007890 La qualité de la rédaction sera prise en compte. Exercice 1. Soit \[ \mathcal{B}=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \] la base canonique de \( \operatorname{Mat}_{2}(\mathbb{R}) \) et soit \( f: \operatorname{Mat}_{2}(\mathbb{R}) \rightarrow \operatorname{Mat}_{2}(\mathbb{R}) \) l'endomorphisme de \( \operatorname{Mat}_{2}(\mathbb{R}) \) tel que, en base canonique, \[ f\left(\left[\begin{array}{ll} x_{1} & x_{2} \\ x_{3} & x_{4} \end{array}\right]\right)=\left(\left[\begin{array}{cc} x_{1}+2 x_{3} & 2 x_{1}-x_{2}+4 x_{3}-2 x_{4} \\ -x_{3} & -2 x_{3}+x_{4} \end{array}\right]\right) \] (a) Montrer que \[ A=\mu_{\mathcal{B}, \mathcal{B}}(f)=\left(\begin{array}{cccc} 1 & 0 & 2 & 0 \\ 2 & -1 & 4 & -2 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right) \] où \( \mu_{\mathcal{B}, \mathcal{B}}(f) \) est la matrice associée à \( f \) dans la base canonique. \( ~ \) trer que l'endomorphisme \( f \) est diagonalisable. Déterminer une base \( \mathcal{B}^{\prime} \) de \( \operatorname{Mat}_{2}(\mathbb{R}) \) formée de vecteurs propres de \( \operatorname{Mat}_{2}(\mathbb{R}) \), la matrice de changement de base \( P:=\mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\mathrm{Mat}_{2}(\mathbb{R})}\right) \) et la matrice diagonale \( D:=\mu_{\mathcal{B}^{\prime}, \mathcal{B}^{\prime}}(f) \) telles que \[ \mu_{\mathcal{B}^{\prime}, \mathcal{B}^{\prime}}(f)=\left(\mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\operatorname{Mat}_{2}(\mathbb{R})}\right)\right)^{-1} \mu_{\mathcal{B}, \mathcal{B}}(f) \mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\operatorname{Mat}_{2}(\mathbb{R})}\right) \] Autrement dit, \[ D=P^{-1} A P \] où \( A=\mu_{\mathcal{B}, \mathcal{B}}(f) \).
Other France Jan 22, 2025
Numéro d'étudiant : La qualité de la rédaction sera prise en compte. Exercice 1. Soit \[ \mathcal{B}=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \] la base canonique de \( \operatorname{Mat}_{2}(\mathbb{R}) \) et soit \( f: \operatorname{Mat}_{2}(\mathbb{R}) \rightarrow \operatorname{Mat}_{2}(\mathbb{R}) \) l'endomorphisme de \( \operatorname{Mat}_{2}(\mathbb{R}) \) tel que, en base canonique, \[ f\left(\left[\begin{array}{ll} x_{1} & x_{2} \\ x_{3} & x_{4} \end{array}\right]\right)=\left(\left[\begin{array}{cc} x_{1}+2 x_{3} & 2 x_{1}-x_{2}+4 x_{3}-2 x_{4} \\ -x_{3} & -2 x_{3}+x_{4} \end{array}\right]\right) \] (a) Montrer que \[ A=\mu_{\mathcal{B}, \mathcal{B}}(f)=\left(\begin{array}{cccc} 1 & 0 & 2 & 0 \\ 2 & -1 & 4 & -2 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right) \] où \( \mu_{\mathcal{B}, \mathcal{B}}(f) \) est la matrice associée à \( f \) dans la base canonique. (b) Déterminer le polynôme caractéristique \( \chi_{f}(x) \). (c) Déterminer les valeurs propres de \( f \), leurs multiplicités algébriques et montrer que l'endomorphisme \( f \) est diagonalisable. (d) Déterminer une base \( \mathcal{B}^{\prime} \) de \( \operatorname{Mat}_{2}(\mathbb{R}) \) formée de vecteurs propres de \( \operatorname{Mat}_{2}(\mathbb{R}) \), la matrice de changement de base \( P:=\mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\operatorname{Mat}_{2}(\mathbb{R})}\right) \) et la matrice diagonale \( D:=\mu_{\mathcal{B}^{\prime}, \mathcal{B}^{\prime}}(f) \) telles que \[ \mu_{\mathcal{B}^{\prime}, \mathcal{B}^{\prime}}(f)=\left(\mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\operatorname{Mat}_{2}(\mathbb{R})}\right)\right)^{-1} \mu_{\mathcal{B}, \mathcal{B}}(f) \mu_{\mathcal{B}^{\prime}, \mathcal{B}}\left(\operatorname{Id}_{\operatorname{Mat}_{2}(\mathbb{R})}\right) \] Autrement dit, \[ D=P^{-1} A P \] où \( A=\mu_{\mathcal{B}, \mathcal{B}}(f) \).
Other France Jan 22, 2025
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