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Other Questions from Jan 22,2025

Browse the Other Q&A Archive for Jan 22,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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32. กลมชนกต้องการถ่ายภาพกมลรัตน์ยืนอยู่กลางแจ้ง มีแสงแดดจ้า เธอต้องการ ปรับแสงถ่ายรูปอย่างไร เพื่อให้ได้ภาพชัดที่สุด 1) ลดขนาดช่องของไดอะแฟรมหรือลดความเร็วชัตเตอร์ 2) ลดขนาดช่องของไดอะแฟรมหรือเพิ่มความเร็วชัดเตอร์ 3) เพิ่มขนาดช่องของไดอะแฟรมหรือเพิ่มความเร็วชัดเตอร์ 4) เพิ่มขนาดช่องของไดอะแฟรมหรือลดความเร็วชัดเตอร์ 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. 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 \( A=\mu_{\mathcal{B}, \mathcal{B}}(f) \). 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 \( A=\mu_{\mathcal{B}, \mathcal{B}}(f) \). 3.2. CASH RECEIPT AND ACCOUNTING EQUATION You are provided with the information relating to Kasi Car wash for September 2022 The business is owned by Mr. J.G. Vilakazi REQUIRED: 3.1 Prepare the Cash receipt journal of Kasi Car wash 2022 . Transactions: September 2022 \( \begin{array}{l}\text { Mr J.G Vilakasi, owner of Kasi Car wash slarted his business with cash } \\ \text { contribution of R10 } 000 \text { as capital; receipt } 001 \text { was issued and money was } \\ \text { deposited in the current account of the business. }\end{array} \) Esercizio 3. È data l'applicazione lineare \( F: \mathbb{R}_{\leq 2}[x] \rightarrow \mathcal{M}_{2}(\mathbb{R}) \) tale che \[ F(p(x))=\left(\begin{array}{cc}p(1) & p(0) \\ p(0) & p(-1)\end{array}\right) \] (a) Senza effettuare calcoli, dire perché esiste almeno un vettore di \( \mathcal{M}_{2}(\mathbb{R}) \) con controimmagine vuota sotto l'azione di \( F \). (b) Si calcoli la matrice associata ad \( F \) rispetto alla base \( \mathcal{B}=\left\{x, x+1, x^{2}-1\right\} \) di \( \mathbb{R}_{\leq 2}[x] \) e alla base canonica di \( \mathcal{M}_{2}(\mathbb{R}) \). (c) Stabilire se \( F \) è iniettiva, suriettiva, invertibile. Esercizio 3. È data l'applicazione lineare \( F: \mathbb{R}_{\leq 2}[x] \rightarrow \mathcal{M}_{2}(\mathbb{R}) \) tale che \[ F(p(x))=\left(\begin{array}{cc}p(1) & p(0) \\ p(0) & p(-1)\end{array}\right) \] (a) Senza effettuare calcoli, dire perché esiste almeno un vettore di \( \mathcal{M}_{2}(\mathbb{R}) \) con controimmagine vuota sotto l'azione di \( F \). (b) Si calcoli la matrice associata ad \( F \) rispetto alla base \( \mathcal{B}=\left\{x, x+1, x^{2}-1\right\} \) di \( \mathbb{R}_{\leq 2}[x] \) e alla base canonica di \( \mathcal{M}_{2}(\mathbb{R}) \). With which character can a user start a formula in Excel? A / B \( \quad \) * C D + What does grouping shapes allow you to do in a presentation software? A rotate the shapes at different angles B rearrange the order of the shapes on the slide C combine multiple shapes into a single object D change the color and size of individual shapes What setting can users set if they want all email with the same sub- ject line to be grouped together? A Group email into conversations. B Show the Folder Pane. C Show the Reading Pane. D Show each message separately.
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