From 11b92baaa83bf129fc142acae2e08fd7990de18b Mon Sep 17 00:00:00 2001 From: Crizomb Date: Thu, 18 Apr 2024 08:00:03 +0200 Subject: [PATCH] * add conversion from pdf to latex-like (.mmd) format with nougat * change vector_db_manager.py to handle .mmd * add "conversion" tab * add math mode checkbox in maintab --- backend/inference.py | 14 +- backend/pdf_to_mmd.py | 16 ++ backend/vector_db_manager.py | 19 +- documents/mmds/2009L1feuille2bis.mmd | 170 ++++++++++++++++++ documents/pdfs/2009L1feuille2bis.pdf | Bin 0 -> 111535 bytes documents/pdfs/The-UK-Constitution.pdf | Bin 0 -> 721403 bytes documents/pdfs/corr_exam.pdf | Bin 276517 -> 0 bytes .../2009L1feuille2bis.mmd/chroma.sqlite3 | Bin 0 -> 786432 bytes .../data_level0.bin | Bin 0 -> 4236000 bytes .../header.bin | Bin 0 -> 100 bytes .../length.bin | Bin 0 -> 4000 bytes .../link_lists.bin | 0 .../The-UK-Constitution.pdf/chroma.sqlite3 | Bin 0 -> 634880 bytes front_end/corr_exam.pdf | Bin 276517 -> 0 bytes .../bc51931986aa4ff95e72/integration.pdf | Bin 907400 -> 0 bytes front_end/flagged/log.csv | 9 - front_end/main.py | 52 ++++-- 17 files changed, 247 insertions(+), 33 deletions(-) create mode 100644 backend/pdf_to_mmd.py create mode 100644 documents/mmds/2009L1feuille2bis.mmd create mode 100644 documents/pdfs/2009L1feuille2bis.pdf create mode 100644 documents/pdfs/The-UK-Constitution.pdf delete mode 100644 documents/pdfs/corr_exam.pdf create mode 100644 documents/vector_db/multilingual-e5-large/2009L1feuille2bis.mmd/chroma.sqlite3 create mode 100644 documents/vector_db/multilingual-e5-large/The-UK-Constitution.pdf/3d8364ae-0b40-4e08-a6dd-ec0d55f4bff2/data_level0.bin create mode 100644 documents/vector_db/multilingual-e5-large/The-UK-Constitution.pdf/3d8364ae-0b40-4e08-a6dd-ec0d55f4bff2/header.bin create mode 100644 documents/vector_db/multilingual-e5-large/The-UK-Constitution.pdf/3d8364ae-0b40-4e08-a6dd-ec0d55f4bff2/length.bin create mode 100644 documents/vector_db/multilingual-e5-large/The-UK-Constitution.pdf/3d8364ae-0b40-4e08-a6dd-ec0d55f4bff2/link_lists.bin create mode 100644 documents/vector_db/multilingual-e5-large/The-UK-Constitution.pdf/chroma.sqlite3 delete mode 100644 front_end/corr_exam.pdf delete mode 100644 front_end/flagged/PDF File/bc51931986aa4ff95e72/integration.pdf delete mode 100644 front_end/flagged/log.csv diff --git a/backend/inference.py b/backend/inference.py index de302e6..fcfd81e 100644 --- a/backend/inference.py +++ b/backend/inference.py @@ -17,28 +17,32 @@ class InferenceInstance: self.nb_chunks_retrieved = nb_chunks_retrieved def get_next_token(self, input_user: str, doc_name: str) -> Iterator[Dict[str, str]]: + is_pdf = doc_name.endswith(".pdf") + print(f"doc_name: {doc_name}") new_assistant_message = {"role": "assistant", "content": ""} search_results = self._get_search_results(input_user, doc_name) print(f"search results: {search_results}") - pages = self._update_history(input_user, search_results) + pages = self._update_history(input_user, search_results, is_pdf) pages_info = f"pages used : p" + " p".join(pages) print(f"history: {self.history}") completion = self._get_completion() for chunk in completion: - new_assistant_message["content"] += chunk.choices[0].delta.content - yield pages_info + " " + new_assistant_message["content"] + if chunk.choices[0].delta.content: + new_assistant_message["content"] += chunk.choices[0].delta.content + yield pages_info + "\n\n " + new_assistant_message["content"] def _get_search_results(self, input_user: str, doc_name: str): print(f"input_user: {input_user}") vector_db = self.vector_db_manager.get_chroma(doc_name) return vector_db.similarity_search(input_user, k=4) - def _update_history(self, input_user: str, search_results): + def _update_history(self, input_user: str, search_results, is_pdf): some_context = "" pages = [] for result in search_results: - pages.append(str(result.metadata['page'])) + if is_pdf: + pages.append(str(result.metadata['page'])) some_context += result.page_content + "\n\n" self.history.append({"role": "system", "content": f"relevant content for user question {some_context}"}) self.history.append({"role": "user", "content": input_user}) diff --git a/backend/pdf_to_mmd.py b/backend/pdf_to_mmd.py new file mode 100644 index 0000000..d697dfb --- /dev/null +++ b/backend/pdf_to_mmd.py @@ -0,0 +1,16 @@ +import subprocess + + +def pdf_to_mmd(path_input: str): + """ + Convert a PDF file to MMD format using the Nougat library + https://github.com/facebookresearch/nougat + + stream stderr to the front end + """ + output_dir = "../documents/mmds" + command = ['nougat', path_input, "-o", output_dir] + subprocess.run(command) + + + diff --git a/backend/vector_db_manager.py b/backend/vector_db_manager.py index 02f7e24..6f8e99e 100644 --- a/backend/vector_db_manager.py +++ b/backend/vector_db_manager.py @@ -14,7 +14,6 @@ class VectorDbManager: self.db_directory = db_directory self.chunk_size = chunk_size - def create_vector_store_from_pdf(self, pdf_path): """ create a chroma vector store from a pdf file path @@ -26,7 +25,7 @@ class VectorDbManager: """ pdf_path = Path(pdf_path) pdf_name = pdf_path.name - vector_directory = self.db_directory/self.embedding_name/pdf_name + vector_directory = self.db_directory / self.embedding_name / pdf_name if os.path.isdir(vector_directory): print(f"{vector_directory} found, not recreating a vector store") @@ -49,7 +48,7 @@ class VectorDbManager: docs = text_splitter.split_documents(docs) vectorstore = Chroma.from_documents(docs, self.embedding_function, persist_directory=vector_directory) - print("vector store created") + print("pdf vector store created") print(vectorstore) def create_vector_store_from_latex(self, latex_path: Path): @@ -62,7 +61,7 @@ class VectorDbManager: :return: """ doc_name = latex_path.name - vector_directory = self.db_directory/self.embedding_name/doc_name + vector_directory = self.db_directory / self.embedding_name / doc_name if os.path.isdir(vector_directory): print(f"{vector_directory} found, not recreating a vector store") @@ -71,10 +70,13 @@ class VectorDbManager: print(f"creating vector store for {vector_directory}") with open(latex_path, mode="r") as file: - text_splitter = RecursiveCharacterTextSplitter(chunk_size=self.chunk_size, chunk_overlap=100) - docs = text_splitter.split_document(file.read()) + text_splitter = RecursiveCharacterTextSplitter.from_language(Language.MARKDOWN, chunk_size=self.chunk_size, chunk_overlap=64) + texts = text_splitter.split_text(file.read()) - vectorstore = Chroma.from_documents(docs, self.embedding_function, persist_directory=vector_directory) + print(texts) + vectorstore = Chroma.from_texts(texts, self.embedding_function, persist_directory=vector_directory) + print("latex vector store created") + print(vectorstore) def get_chroma(self, doc_name): """ @@ -83,6 +85,5 @@ class VectorDbManager: :param doc_name: :return: """ - vector_directory = self.db_directory/self.embedding_name/doc_name + vector_directory = self.db_directory / self.embedding_name / doc_name return Chroma(persist_directory=vector_directory, embedding_function=self.embedding_function) - diff --git a/documents/mmds/2009L1feuille2bis.mmd b/documents/mmds/2009L1feuille2bis.mmd new file mode 100644 index 0000000..db35254 --- /dev/null +++ b/documents/mmds/2009L1feuille2bis.mmd @@ -0,0 +1,170 @@ +**Exercices corriges** + +**Algebre lineaire 1** + +## 1 Enonces + +**Exercice 1**: On rappelle que \((E,+,\cdot)\) est un \(\mathbb{K}\)-espace vectoriel si + +1. \((E,+)\) est un groupe commutatif\(\,\); +2. \(\forall x,y\in E,\,\forall\alpha\in\mathbb{K},\,\alpha\cdot(x+y)=\alpha\cdot x +\alpha\cdot y\,\); +3. \(\forall x\in E,\,\forall\alpha,\beta\in\mathbb{K},\,(\alpha+\beta)\cdot x= \alpha\cdot x+\beta\cdot x\,\); +4. \(\forall x\in E,\,\forall\alpha,\beta\in\mathbb{K},\,\alpha\cdot(\beta\cdot x )=(\alpha\beta)\cdot x\,\); +5. \(1\cdot x=x\). + +Soit \((E,+,\cdot)\) un \(\mathbb{K}\)-espace vectoriel. On note \(0_{E}\) l'element neutre de \((E,+)\) (que l'on appelle aussi l'origine de \((E,+,\cdot)\)) et \(0_{\mathbb{K}}\) le nombre zero (dans \(\mathbb{K}\)). Pour tout \(x\) dans \(E\), le symetrique de \(x\) est note \(-x\). + +1. Montrer que, pour tout \(x\in E\), \(x+x=2\cdot x\). +2. Montrer que, pour tout \(x\in E\), \(0_{\mathbb{K}}\cdot x=0_{E}\). +3. Montrer que, pour tout \(x\in E\), \((-1)\cdot x=-x\). + +**Exercice 2**: Soient \(F_{1},\ldots,F_{m}\) des sous-espaces vectoriels d'un \(\mathbb{R}\)-espace vectoriel \((E,+,\cdot)\). Montrer que \(F:=F_{1}\cap\ldots\cap F_{m}\) est un sous-espace vectoriel de \(E\). + +**Exercice 3**: Soient \((E,+,\cdot)\) un \(\mathbb{R}\)-espace vectoriel, \(\{x_{1},\ldots,x_{m}\}\) une famille de vecteurs de \(E\). Montrer que \(F:=\operatorname{vect}\{x_{1},\ldots,x_{m}\}\) est un sous-espace vectoriel de \(E\). + +**Exercice 4**: Soient \((E,+,\cdot)\) un \(\mathbb{R}\)-espace vectoriel, \(F\) un sous-espace vectoriel de \(E\) et \(A,B\) deux sous-ensembles de \(E\). + +1. Montrer que, si \(A\subset B\), alors \(\operatorname{vect}A\subset\operatorname{vect}B\). +2. Montrer que \(A\) est un sous-espace vectoriel de \(E\) si et seulement si \(\operatorname{vect}A=A\). +3. Montrer que, si \(A\subset B\subset F\) et \(A\) engendre \(F\), alors \(B\) engendre \(F\). + +**Exercice 5**: Considerons les vecteurs de \(\mathbb{R}^{4}\) suivants : + +\[\mathbf{e}_{1}=\left(\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right),\quad\mathbf{e}_{2}=\left(\begin{array}{c}0\\ 1\\ 2\\ -1\end{array}\right),\quad\mathbf{e}_{3}=\left(\begin{array}{c}1\\ 0\\ -2\\ 3\end{array}\right),\quad\mathbf{e}_{4}=\left(\begin{array}{c}2\\ 1\\ 0\\ -1\end{array}\right).\] + +La famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\) est-elle libre\(\,\)? Est-ce une base de \(\mathbb{R}^{4}\,\)? + +**Exercice 6**: Considerons les vecteurs de \(\mathbb{R}^{4}\) suivants : + +\[\mathbf{e}_{1}=\left(\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right),\quad\mathbf{e}_{2}=\left(\begin{array}{c}0\\ 1\\ 2\\ 1\end{array}\right),\quad\mathbf{e}_{3}=\left(\begin{array}{c}1\\ 0\\ -2\\ 3\end{array}\right),\quad\mathbf{e}_{4}=\left(\begin{array}{c}1\\ 1\\ 2\\ -2\end{array}\right).\]1. La famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\) est-elle libre? +2. Quel est le rang de la famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\)? +3. Determiner une relation entre les nombres reels \(\alpha\) et \(\beta\) pour que le vecteur \(\mathbf{u}=(1,1,\alpha,\beta)^{t}\) appartienne au sous-espace vectoriel engendre par la famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\). + +**Exercice 7**: Soit \(E=\mathbb{R}^{\mathbb{R}}\), l'espace des fonctions de \(\mathbb{R}\) dans \(\mathbb{R}\). + +1. Soient \(c\) et \(s\) les fonctions definies par \[\forall x\in\mathbb{R},\quad c(x)=\cos x\quad\text{et}\quad s(x)=\sin x.\] Montrer que \(\{c,s\}\) est une famille libre de \(E\). Quelle est la dimension du sous-espace vectoriel \(T\) engendre par la famille \(\{c,s\}\)? +2. Soient \(\alpha,\beta,\gamma\) trois reels fixes. Soient \(f,g,h\) les fonctions definies par \[\forall x\in\mathbb{R},\quad f(x)=\cos(x+\alpha),\quad g(x)=\cos(x+\beta) \quad\text{et}\quad h(x)=\cos(x+\gamma).\] Montrer que \(f,g,h\) appartiennent a \(T\), et expliciter leurs coordonnees dans la base \(\{c,s\}\) de \(T\). La famille \(\{f,g,h\}\) est-elle libre? Quel est son rang? +3. Soient \(a_{1},a_{2},a_{3}\) trois reels distincts. Pour tout entier \(k\in\{1,2,3\}\) on note \(f_{k}\) la fonction definie sur \(\mathbb{R}\) par \[\forall x\in\mathbb{R},\quad f_{k}(x)=\left|x-a_{k}\right|.\] Montrer que \(\{f_{1},f_{2},f_{3}\}\) est une famille libre de \(E\). + +**Exercice 8**: +1. On rappelle que \(\mathcal{C}_{0}(\mathbb{R})\) designe l'espace des fonctions continues de \(\mathbb{R}\) dans \(\mathbb{R}\). Montrer que \(\mathcal{A}:=\{f\in\mathcal{C}_{0}(\mathbb{R})|\forall x\in\mathbb{R},\;f(x)= f(-x)\}\) et \(\mathcal{B}:=\{f\in\mathcal{C}_{0}(\mathbb{R})|\forall x\in\mathbb{R},\;f(x)= -f(-x)\}\) sont des sous-espaces vectoriels de \(\mathcal{C}_{0}(\mathbb{R})\). Sont-ils en somme directe? +2. Montrer que \(A:=\{(x,y,z)\in\mathbb{R}^{3}|x+y+z=0\}\) et \(B:=\{(x,y,z)\in\mathbb{R}^{3}|x-y+z=0\}\) sont des sous-espaces vectoriels de \(\mathbb{R}^{3}\). Sont-ils en somme directe? + +**Exercice 9**: +1. Soient \(F:=\{(x,x,x)\in\mathbb{R}^{3}|x\in\mathbb{R}\}\) et \(G:=\{(0,y,z)\in\mathbb{R}^{3}|y,z\in\mathbb{R}\}\). Montrer que \(F\) et \(G\) sont deux sous-espaces vectoriels de \(\mathbb{R}^{3}\). Preciser leurs bases et leurs dimensions. Sont-ils en somme directe? +2. Soit \(H:=\{(x,y,z,t)\in\mathbb{R}^{4}|x=2y-z,\;t=x+y+z\}\). Verifier que \(H\) est un sous-espace vectoriel de \(\mathbb{R}^{4}\). En donner une base et la dimension. + +**Exercice 10**: Soient \((E,+,\cdot)\) un \(\mathbb{R}\)-espace vectoriel et \(A,B,C\) trois sous-espaces vectoriels de \(E\). + +1. Montrer que \((A\cap C)+(B\cap C)\subset(A+B)\cap C\). Donner un exemple dans \(\mathbb{R}^{2}\) pour lequel l'inclusion est stricte. +2. Montrer que, si \(A+B=A+C\), \(A\cap B=A\cap C\) et \(B\subset C\), alors \(B=C\). + +**Exercice 11**: On considere l'application donnee par + +1. Montrer que \(\varphi\) est une application lineaire. Determiner l'image par \(\varphi\) des vecteurs de la base canonique \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) de \(\mathbb{R}^{3}\). Calculer \(\varphi(2\mathbf{e}_{1}+\mathbf{e}_{2}-\mathbf{e}_{3})\). +2. Determiner le noyau de \(\varphi\). En donner une base et preciser sa dimension. + +3. L'application \(\varphi\) est-elle injective? surjective? bijective? +4. Soit \(\psi\) l'application lineaire donnee par \[\psi\colon \mathbb{R}^{2} \longrightarrow \mathbb{R}^{3}\] \[\left(\begin{array}{c}x\\ y\end{array}\right) \longmapsto \left(\begin{array}{c}x-y\\ x+y\\ x+2y\end{array}\right).\] Determiner \(\varphi\circ\psi\). + +**Exercice 12**: On considere l'application donnee par + +\[\varphi\colon \mathbb{R}^{3} \longrightarrow \mathbb{R}^{2}\] \[\left(\begin{array}{c}x\\ y\\ z\end{array}\right) \longmapsto \left(\begin{array}{c}y+z\\ x\end{array}\right)\] + +ainsi que les vecteurs \(\mathbf{u}:=(1,2,3)^{t}\) et \(\mathbf{v}:=(1,1,1)^{t}\). + +1. Montrer que \(\varphi\) est lineaire. Determiner \(\varphi(\mathbf{u})\), \(\varphi(\mathbf{v})\) et \(\varphi(\mathbf{u}-2\mathbf{v})\). +2. Determiner le noyau de \(\varphi\). En donner une base et preciser sa dimension. +3. Determiner l'image de \(\varphi\). En donner une base et preciser sa dimension. + +**Exercice 13**: Soient \(E\) et \(F\) deux \(\mathbb{R}\)-espaces vectoriels et \(\varphi\) une application lineaire de \(E\) dans \(F\). Soit \(\mathcal{A}:=\{x_{1},\ldots,x_{m}\}\) une famille de vecteurs de \(E\). + +1. Montrer que, si \(\mathcal{A}\) est liee, alors \(f(\mathcal{A})=\{\varphi(x_{1}),\ldots,\varphi(x_{m})\}\) est liee. +2. Montrer que, si \(\varphi(\mathcal{A})\) est libre, alors \(\mathcal{A}\) est libre. +3. Montrer que, si \(\mathcal{A}\) est libre et \(\varphi\) est injective, alors \(\varphi(\mathcal{A})\) est libre. + +## 2 Solutions + +**Solution de l'exercice 1** + +1. Pour tout \(x\in E\), \(2\cdot x=(1+1)\cdot x=1\cdot x+1\cdot x=x+x\), ou l'on a utilise successivement les axiomes (II-2) et (II-4). +2. On a : \[\begin{array}{rcl}0_{\mathbb{K}}\cdot x&=&(0_{\mathbb{K}}2)\cdot x\\ &=&0_{\mathbb{K}}\cdot(2\cdot x)&\text{[d'apres l'axiome (II-3)]}\\ &=&0_{\mathbb{K}}\cdot(x+x)&\text{[d'apres la question (1)]}\\ &=&0_{\mathbb{K}}\cdot x+0_{\mathbb{K}}\cdot x.\end{array}\] En simplifiant (c'est-a-dire, en ajoutant \(-(0_{\mathbb{K}}\cdot x)\) des deux cotes), on obtient l'egalite \(0_{E}=0_{\mathbb{K}}\cdot x\). +3. D'apres la question (2), \(0_{E}=0_{\mathbb{K}}\cdot x=(1+(-1))\cdot x=(1\cdot x)+((-1)\cdot x)=x+((-1) \cdot x)\), ou la troisieme egalite resulte de l'axiome (II-2) et ou la derniere egalite resulte de l'axiome (II-4). On en deduit que \((-1)\cdot x\) est le symetrique de \(x\), c'est-a-dire, \(-x\). + +**Solution de l'exercice 2**: Nous devons montrer que pour tous \(x,y\in F\) et pour tout \(\alpha\in\mathbb{R}\), \(x+\alpha y\in F\). Soient donc \(x,y\in F\) et \(\alpha\in\mathbb{R}\) quelconques. Par definition de l'intersection, pour tout \(k\in\{1,\ldots,m\}\), \(x,y\in F_{k}\). Comme \(F_{k}\) est un sous-espace vectoriel de \(E\) nous deduisons que + +\[x+\alpha y\in F_{k},\]et ce pour tout \(k\in\{1,\ldots,m\}\). Donc \(x+\alpha y\) appartient a l'intersection des \(F_{k}\), c'est-a-dire, a \(F\). + +**Solution de l'exercice 3** : Remarquons tout d'abord que \(F\) est non vide, puisque que + +\[0_{E}=0\cdot x_{1}+\cdots+0\cdot x_{m}\in F.\] + +Soient \(x,y\in F\) et \(\alpha\in\mathbb{R}\) quelconques. Alors \(x\) et \(y\) s'ecrivent + +\[x=\alpha_{1}x_{1}+\cdots+\alpha_{m}x_{m}\quad\mbox{et}\quad y=\beta_{1}x_{1}+ \cdots+\beta_{m}x_{m},\] + +avec \(\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{m}\in\mathbb{R}\). Donc, + +\[x+\alpha y = (\alpha_{1}x_{1}+\cdots+\alpha_{m}x_{m})+\alpha(\beta_{1}x_{1}+ \cdots+\beta_{m}x_{m})\] \[= (\alpha_{1}+\alpha\beta_{1})x_{1}+\cdots+(\alpha_{m}+\alpha\beta _{m})x_{m}.\] + +Par consequent, \(x+\alpha y\) est une combinaison lineaire des vecteurs \(x_{1},\ldots,x_{m}\), c'est-a-dire, un element de \(F\). + +**Solution de l'exercice 4** : +1. Supposons que \(A\subset B\), et montrons que tout element de \(\mbox{vect}\,A\) appartient a vect \(B\). Soit donc \(x\) quelconque dans \(\mbox{vect}\,A=\emptyset\), alors \(\mbox{vect}\,A=\{0\}\) et donc \(x\) est forcement le vecteur nul. Comme \(\mbox{vect}\,B\) est un sous-espace vectoriel, \(\mbox{vect}\,B\ni 0\) et l'on a bien \(\mbox{vect}\,A\subset\mbox{vect}\,B\). Si \(A\) est non vide, alors \[\exists p\in\mathbb{N}^{*},\ \exists x_{1},\ldots,x_{p}\in A,\ \exists\alpha_{1},\ldots, \alpha_{p}\in\mathbb{R}\colon\quad x=\alpha_{1}x_{1}+\cdots+\alpha_{p}x_{p}.\] Puisque \(A\subset B\), les \(x_{k}\) sont aussi dans \(B\), de sorte que \(x\) est une combinaison lineaire de vecteurs de \(B\), c'est-a-dire, un element de \(\mbox{vect}\,B\). On a donc encore \(\mbox{vect}\,A\subset\mbox{vect}\,B\). +2. Supposons que \(A=\mbox{vect}\,A\). Puisque \(\mbox{vect}\,A\) est un sous-espace vectoriel, il en est de meme de \(A\). Reciproquement, supposons que \(A\) soit un sous-espace vectoriel, et montrons que \(A=\mbox{vect}\,A\). Remarquons que tout element de \(A\) est une combinaison lineaire particuliere d'elements de \(A\) (prendre \(p=1\), \(\alpha_{1}=1\) et \(x_{1}=x\)). Donc on a clairement l'inclusion \(A\subset\mbox{vect}\,A\). De plus, si \(A\) est un sous-espace vectoriel, alors \(A\) est non vide. Soit alors \(x\in\mbox{vect}\,A\) : \[\exists p\in\mathbb{N}^{*},\ \exists x_{1},\ldots,x_{p}\in A,\ \exists\alpha_{1},\ldots,\alpha_{p}\in\mathbb{R}\colon\quad x= \alpha_{1}x_{1}+\cdots+\alpha_{p}x_{p}.\] Puisque \(A\) est stable par combinaison lineaire, \(x\in A\). On a donc aussi l'inclusion \(\mbox{vect}\,A\subset A\). +3. D'apres le point (1), \(\mbox{vect}\,A\subset\mbox{vect}\,B\subset\mbox{vect}\,F\). Or, \(\mbox{vect}\,F=F\) puisque \(F\) est un sous-espace vectoriel. De plus, \(\mbox{vect}\,A=F\) puisque \(A\) engendre \(F\). Finalement, on a : \[F\subset\mbox{vect}\,B\subset F,\] ce qui montre que \(\mbox{vect}\,B=F\). Autrement dit, \(B\) engendre \(F\). + +**Solution de l'exercice 5** : On resout l'equation vectorielle \(\alpha\mathbf{e}_{1}+\beta\mathbf{e}_{2}+\gamma\mathbf{e}_{3}+\delta\mathbf{e}_ {4}=\mathbf{0}\). Ceci revient resoudre le systeme lineaire + +\[\left\{\begin{array}{rcl}0&=&\alpha+\gamma+2\delta,\\ 0&=&\alpha+\beta+\delta,\\ 0&=&\alpha+2\beta-2\gamma,\\ 0&=&\alpha-\beta+3\gamma-\delta.\end{array}\right.\] + +On trouve que la seule solution possible est \(\alpha=\beta=\gamma=\delta=0\). Donc la famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\) est libre, et puisque son cardinal est egal a la dimension de \(\mathbb{R}^{4}\), c'est une base de \(\mathbb{R}^{4}\). + +**Solution de l'exercice 6**: + +1. On resout l'equation vectorielle \(\alpha\mathbf{e}_{1}+\beta\mathbf{e}_{2}+\gamma\mathbf{e}_{3}+\delta\mathbf{e }_{4}=\mathbf{0}.\) Ceci revient resoudre le systeme lineaire \[\left\{\begin{array}{rcl}0&=&\alpha+\gamma+\delta,\\ 0&=&\alpha+\beta+\delta,\\ 0&=&\alpha+2\beta-2\gamma+2\delta,\\ 0&=&\alpha+\beta+3\gamma-2\delta.\end{array}\right.\] On trouve que ce systeme est equivalent au systeme \[\left\{\begin{array}{rcl}0&=&\alpha+\gamma+\delta,\\ 0&=&\beta-\gamma,\\ 0&=&\gamma-\delta.\end{array}\right.\] Ce systeme admet d'autres solutions que la solution nulle. On en deduit que \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\) n'est pas libre. +2. D'apres ce qui precede, le rang de la famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\) est inferieur ou egal a 3. On considere alors la famille \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}.\) On verifie facilement qu'elle est libre, de sorte que le rang cherche est en fait egal a 3. +3. Pour que \(\mathbf{u}\) appartienne au _sev_ engendre par \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}\), il faut que l'equation vectorielle \[\mathbf{u}=\alpha\mathbf{e}_{1}+\beta\mathbf{e}_{2}+\gamma\mathbf{e}_{3}+ \delta\mathbf{e}_{4}\] admette au moins une solution. On cherche donc a resoudre le systeme lineaire \[\left\{\begin{array}{rcl}1&=&\alpha+\gamma+\delta,\\ 1&=&\alpha+\beta+\delta,\\ a&=&\alpha+2\beta-2\gamma+2\delta,\\ b&=&\alpha+\beta+3\gamma-2\delta.\end{array}\right.\] On verifie que ce systeme est equivalent au systeme \[\left\{\begin{array}{rcl}1&=&\alpha+\gamma+\delta,\\ 0&=&\beta-\gamma,\\ a-1&=&-\gamma+\delta,\\ b-1&=&3\gamma-3\delta.\end{array}\right.\] En considerant les deux dernieres equations, on voit que le systeme n'a de solution que si \(b-1=-3(a-1)\), c'est-a-dire, si \(b+3a=4.\) + +**Solution de l'exercice 7**: + +1. Considerons l'equation \(\alpha c+\beta s=0\) dans \(\mathbb{R}^{\mathbb{R}}.\) Cette equation est equivalente a \[\forall x\in\mathbb{R},\quad\alpha\cos x+\beta\sin x=0.\] Les choix \(x=0\) et \(x=\pi/2\) donnent respectivement \(\alpha=0\) et \(\beta=0.\) La famille \(\{c,s\}\) est donc libre, et la dimension de \(T\) est egale a 2. +2. Puisque \(\cos(x+\alpha)=\cos x\cos\alpha-\sin x\sin\alpha,\) on voit que \[f=\cos\alpha\cdot c-\sin\alpha\cdot s\in T\] et que les coordonnees de \(f\) dans la base \(\{c,s\}\) de \(T\) sont donnees par le couple \((\cos\alpha,-\sin\alpha).\) De meme, \[g=\cos\beta\cdot c-\sin\beta\cdot s\in T\quad\mbox{et}\quad h=\cos\gamma \cdot c-\sin\gamma\cdot s\in T;\]les coordonnees de \(g\) et \(h\) dans la base \(\{c,s\}\) de \(T\) sont donnees respectivement par les couples \((\cos\beta,-\sin\beta)\) et \((\cos\gamma,-\sin\gamma)\). La famille \(\{f,g,h\}\) ne peut pas etre libre, puisque son cardinal est egal a 3 alors que la dimension de l'espace vectoriel \(T\) est egale a 2. Son rang vaut au plus 2 (car \(\dim T=2\)) et au moins 1 (car les fonctions \(f,g,h\) sont non nulles). Le rang est egal a 1 lorsque \(f,g,h\) sont colineaires, c'est-a dire lorsqu'il existe \(a\) et \(b\) dans \(\mathbb{R}\) tels que \(f=ag=bh\) ou, de maniere equivalente, lorsque \[\left(\begin{array}{c}\cos\alpha\\ -\sin\alpha\end{array}\right)=a\left(\begin{array}{c}\cos\beta\\ -\sin\beta\end{array}\right)=b\left(\begin{array}{c}\cos\gamma\\ -\sin\gamma\end{array}\right).\] Des equations \(\cos\alpha=a\cos\beta\) et \(\sin\alpha=a\sin\beta\) on tire, en les elevant au carre et en les sommant, que \(a^{2}=1\), c'est-a-dire, que \(a\in\{-1,1\}\). Si \(a=1\), alors \(\beta=\alpha+2k\pi\), et si \(a=-1\), alors \(\beta=\alpha+\pi+2k\pi\). En resume, \(f\) et \(g\) sont colineaires si et seulement si \(\beta\in\{\alpha\}+\pi\mathbb{Z}\). De meme, \(f\) et \(h\) sont colineaires si et seulement si \(\gamma\in\{\alpha\}+\pi\mathbb{Z}\). La famille \(\{f,g,h\}\) est donc de rang 1 lorsque \(\alpha\), \(\beta\) et \(\gamma\) different d'un multiple entier de \(\pi\) ; elle est de rang 2 dans le cas contraire. +3. Considerons l'equation \(\alpha f_{1}+\beta f_{2}+\gamma f_{3}=0\) dans \(\mathbb{R}^{\mathbb{R}}\), qui equivaut a la condition \[\forall x\in\mathbb{R},\quad\alpha f_{1}(x)+\beta f_{2}(x)+\gamma f_{3}(x)=0.\] Les choix \(x=a_{1}\), \(x=a_{2}\) et \(x=a_{3}\) donnent respectivement les equations \[\beta\left|a_{1}-a_{2}\right|+\gamma\left|a_{1}-a_{3}\right| = 0,\] \[\alpha\left|a_{2}-a_{1}\right|+\gamma\left|a_{2}-a_{3}\right| = 0,\] \[\alpha\left|a_{3}-a_{1}\right|+\beta\left|a_{3}-a_{2}\right| = 0.\] Posons \(a:=\left|a_{3}-a_{1}\right|\), \(b:=\left|a_{3}-a_{2}\right|\) et \(c:=\left|a_{1}-a_{2}\right|\). Le systeme d'equations precedent s'ecrit \[\left\{\begin{array}{ccl}0&=&a\alpha+b\beta,\\ 0&=&c\alpha+b\gamma,\\ 0&=&c\beta+a\gamma.\end{array}\right.\] En resolvant ce systeme lineaire, et en tenant compte du fait que \(a\), \(b\) et \(c\) sont non nuls, on voit que la seule solution possible est \(\alpha=\beta=\gamma=0\). On peut aussi ecrire le systeme sous forme matricielle, et remarquer, pour arriver a la meme conclusion, que la matrice \[\left[\begin{array}{ccc}a&b&0\\ c&0&b\\ 0&c&a\end{array}\right]\] a pour determinant le reel non nul \(-2abc\). + +**Solution de l'exercice 8**: + +1. La fonction nulle \(\nu\) (definie par \(\nu(x)=0\) pour tout \(x\in\mathbb{R}\)) appartient a \(\mathcal{A}\) et a \(\mathcal{B}\). Donc, \(\mathcal{A}\) et \(\mathcal{B}\) sont non vides. De plus, pour toutes fonctions \(f,g\in\mathcal{A}\) et tout reel \(\alpha\), la fonction \(f+\alpha g\) satisfait : \[\forall x\in\mathbb{R},\quad(f+\alpha g)(x)=f(x)+\alpha g(x)=f(-x)+\alpha g(-x)=( f+\alpha g)(-x).\] Par consequent, \(f+\alpha g\in\mathcal{A}\). Donc \(\mathcal{A}\) est un sous-espace vetoriel de \(\mathcal{C}_{0}(\mathbb{R})\). De meme, pour toutes fonctions \(f,g\in\mathcal{B}\) et tout reel \(\alpha\), la fonction \(f+\alpha g\) satisfait : \[\forall x\in\mathbb{R},\quad(f+\alpha g)(x)=f(x)+\alpha g(x)=-f(-x)-\alpha g(-x)=-( f+\alpha g)(-x).\]Par consequent, \(f+\alpha g\in\mathcal{A}\). Donc \(\mathcal{B}\) est un sous-espace vetoriel de \(\mathcal{C}_{0}(\mathbb{R})\). Soit maintenant \(f\) une fonction de \(\mathcal{A}\cap\mathcal{B}\). Alors, pour tout \(x\in\mathbb{R}\), \[f(x)=f(-x)\quad\mbox{et}\quad f(x)=-f(-x),\] ce qui montre que \(f(x)=0\). Donc \(f=\nu\). On en deduit que \(\mathcal{A}\cap\mathcal{B}=\{\nu\}=\{0_{\mathcal{C}_{0}(\mathbb{R})}\}\), et que \(\mathcal{A}\) et \(\mathcal{B}\) sont en somme directe. +2. Il est facile de voir que \(A\) et \(B\) contiennent le vecteur \(\mbox{nul}\ (0,0,0)\). De plus, si \((x,y,z)\) et \((x^{\prime},y^{\prime},z^{\prime})\) appartiennent a \(A\) et \(\alpha\in\mathbb{R}\), alors \((x,y,z)+\alpha(x^{\prime},y^{\prime},z^{\prime})=(x+\alpha x^{\prime},y+\alpha y ^{\prime},z+\alpha z^{\prime})\) satisfait \[(x+\alpha x^{\prime})+(y+\alpha y^{\prime})+(z+\alpha z^{\prime})=(x+y+z)+ \alpha(x^{\prime}+y^{\prime}+z^{\prime})=0.\] Donc \((x,y,z)+\alpha(x^{\prime},y^{\prime},z^{\prime})\in A\), et \(A\) est un sous-espace vectoriel de \(\mathbb{R}^{3}\). De meme, si \((x,y,z)\) et \((x^{\prime},y^{\prime},z^{\prime})\) appartiennent a \(B\) et \(\alpha\in\mathbb{R}\), alors \((x,y,z)+\alpha(x^{\prime},y^{\prime},z^{\prime})=(x+\alpha x^{\prime},y+\alpha y ^{\prime},z+\alpha z^{\prime})\) satisfait \[(x+\alpha x^{\prime})-(y+\alpha y^{\prime})+(z+\alpha z^{\prime})=(x-y+z)+ \alpha(x^{\prime}-y^{\prime}+z^{\prime})=0.\] Donc \((x,y,z)+\alpha(x^{\prime},y^{\prime},z^{\prime})\in B\), et \(B\) est un sous-espace vectoriel de \(\mathbb{R}^{3}\). Soit maintenant \((x,y,z)\) un vecteur de \(A\cap B\). Alors, \[x+y+z=0\quad\mbox{et}\quad x-y+z=0.\] Le vecteur \((1,0,-1)\) satisfait les deux equations ci-dessus. On voit donc que \(A\cap B\) n'est pas reduit a \(\{(0,0,0)\}\). Les sous-espaces \(A\) et \(B\) ne sont pas en somme directe. + +**Solution de l'exercice 9**: +1. Il est facile de voir que le vecteur \((0,0,0)\) appartient a \(F\) et a \(G\). Donc \(F\) et \(G\) sont non vides. Soient \((x,x,x),(y,y,y)\in F\) et \(\alpha\in\mathbb{R}\). Alors \[(x,x,x)+\alpha(y,y,y)=(x+\alpha y,x+\alpha y,x+\alpha y)\in F.\] Donc \(F\) est un sous-espace vectoriel de \(\mathbb{R}^{3}\). Soient \((0,y,z),(0,y^{\prime},z^{\prime})\in G\) et \(\alpha\in\mathbb{R}\). Alors \[(0,y,z)+\alpha(0,y^{\prime},z^{\prime})=(0,y+\alpha y^{\prime},z+\alpha z^{ \prime})\in G.\] Donc \(G\) est un sous-espace vectoriel de \(\mathbb{R}^{3}\). On voit que \[F=\{x(1,1,1)|x\in\mathbb{R}\}=\mbox{vect}\{(1,1,1)\},\] \[G=\{y(0,1,0)+z(0,0,1)|x,y\in\mathbb{R}\}=\mbox{vect}\{(0,1,0),(0,0,1)\}.\] De plus, on verifie facilement que les familles \(\{(1,1,1)\}\) et \(\{(0,1,0),(0,0,1)\}\) sont libres. Elles forment donc des bases respectives de \(F\) et \(G\). On en deduit que \(\mbox{dim}\,F=1\) et \(\mbox{dim}\,G=2\). Enfin, si \((x,y,z)\in F\cap G\), alors \(x=y=z\) et \(x=0\). Donc \(F\cap G=\{(0,0,0)\}\), et \(F\) et \(G\) sont en somme directe. +2. On verifie facilement que \((0,0,0,0)\in H\), de sorte que \(F\neq\emptyset\). Soient \((x,y,z,t),(x^{\prime},y^{\prime},z^{\prime},t^{\prime})\in H\) et \(\alpha\in\mathbb{R}\). Alors, \((x,y,z,t)+\alpha(x^{\prime},y^{\prime},z^{\prime},t^{\prime})=(x+\alpha x^{ \prime},y+\alpha y^{\prime},z+\alpha z^{\prime},t+\alpha t^{\prime})\) satisfait : \[x+\alpha x^{\prime}=2y-z+\alpha(2y^{\prime}-z^{\prime})=2(y+\alpha y^{\prime})-( z+\alpha z^{\prime}),\] \[t+\alpha t^{\prime}=x+y+z+\alpha(x^{\prime}+y^{\prime}+z^{\prime})=(x+\alpha x^{\prime})+(y+\alpha y ^{\prime})+(z+\alpha z^{\prime}),\] ce qui montre que \((x,y,z,t)+\alpha(x^{\prime},y^{\prime},z^{\prime},t^{\prime})\in H\). Donc \(H\) est un sous-espace vectoriel de \(\mathbb{R}^{4}\). De plus, \[H = \{(2y-z,y,z,x+y+z)|x,y,z\in\mathbb{R}\}\] \[= \{x(0,0,0,1)+y(2,1,0,1)+z(-1,0,1,1)|x,y,z\in\mathbb{R}\}\] \[= \mbox{vect}\{(0,0,0,1),(2,1,0,1),(-1,0,1,1)\}.\]Considerons l'equation vectorielle \(\alpha(0,0,0,1)+\beta(2,1,0,1)+\gamma(-1,0,1,1)=(0,0,0,0)\). Cette equation equivaut au systeme \[\left\{\begin{array}{rcl}0&=&2\beta+\gamma\\ 0&=&2\beta\\ 0&=&\gamma\\ 0&=&\alpha+\beta+\gamma\end{array}\right.\] dont l'unique solution est \(\alpha=\beta=\gamma=0\). La famille \(\{(0,0,0,1),(2,1,0,1),(-1,0,1,1)\}\) est donc libre, et c'est une base de \(H\). + +**Solution de l'exercice 10**: +1. Soit \(x\in(A\cap C)+(B\cap C)\). Alors \(x=a+b\) avec \(a\in A\cap C\) et \(b\in B\cap C\). Puisque \(a\in C\) et \(b\in C\) et \(C\) est un sev, \(a+b\in C\). Donc \(x\) appartient a \(A+B\) et a \(C\). Dans \(\mathbb{R}^{2}\), Si l'on prend \(A=\mbox{vect}\{e_{1}\}\), \(B=\mbox{vect}\{e_{2}\}\) et \(C=\mbox{vect}\{e_{1}+e_{2}\}\), ou \(\{e_{1},e_{2}\}\) est la base canonique, alors \[(A\cap C)+(B\cap C)=\{0\}\cap\{0\}=\{0\}\quad\mbox{et}\quad(A+B)\cap C=\mathbb{ R}^{2}\cap C=C.\] +2. Puisque \(B\subset C\), il suffit de montrer que \(C\subset B\). Soit donc \(x\in C\). Puisque \(0_{E}\in A\), \(x=0_{E}+x\in A+C\). Puisque \(A+C=A+B\), on peut ecrire \(x=a+b\) avec \(a\in A\) et \(b\in B\). Maintenant, \(a=x-b\), ou \(x\in C\) et \(x\in B\subset C\), et puisque \(C\) est un sev, \(a\in C\). Donc \(a\in A\cap C=B\cap C\). Donc \(a\in B\). Finalement, \(x=a+b\) avec \(a\in B\) et \(b\in B\). Puisque \(B\) est un sev, \(x\in B\). + +**Solution de l'exercice 11**: +1. Verifions que \(\varphi\) est lineaire : \[\varphi\left(\alpha\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\beta\left(\begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array}\right)\right)\] \[= \varphi\left(\begin{array}{c}\alpha x+\beta x^{\prime}\\ \alpha y+\beta y^{\prime}\\ \alpha z+\beta z^{\prime}\end{array}\right)\] \[= \left(\begin{array}{c}-(\alpha x+\beta x^{\prime})+2(\alpha y+ \beta y^{\prime})+2(\alpha z+\beta z^{\prime})\\ -8(\alpha x+\beta x^{\prime})+7(\alpha y+\beta y^{\prime})+4(\alpha z+\beta z^ {\prime})\\ -13(\alpha x+\beta x^{\prime})+5(\alpha y+\beta y^{\prime})+8(\alpha z+\beta z ^{\prime})\end{array}\right)\] \[= \alpha\left(\begin{array}{c}-x+2y+2z\\ -8x+7y+4z\\ -13x+5y+8z\end{array}\right)+\beta\left(\begin{array}{c}-x^{\prime}+2y^{ \prime}+2z^{\prime}\\ -8x^{\prime}+7y^{\prime}+4z^{\prime}\\ -13x^{\prime}+5y^{\prime}+8z^{\prime}\end{array}\right)\] \[= \alpha\varphi\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\beta\varphi\left(\begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array}\right).\] + +Ensuite, + +\[\varphi(\mathbf{e}_{1})=\left(\begin{array}{c}-1\\ -8\\ -13\end{array}\right),\quad\varphi(\mathbf{e}_{2})=\left(\begin{array}{c}2\\ 7\\ 5\end{array}\right),\quad\varphi(\mathbf{e}_{3})=\left(\begin{array}{c}2\\ 4\\ 8\end{array}\right).\] + +Enfin, \(2\mathbf{e}_{1}+\mathbf{e}_{2}-\mathbf{e}_{3}=(2,1,-1)^{t}\), de sorte que + +\[\varphi(2\mathbf{e}_{1}+\mathbf{e}_{2}-\mathbf{e}_{3}) = 2\varphi(\mathbf{e}_{1})+\varphi(\mathbf{e}_{2})-\varphi(\mathbf{e }_{3})\] \[= 2\left(\begin{array}{c}-1\\ -8\\ -13\end{array}\right)+\left(\begin{array}{c}2\\ 7\\ 5\end{array}\right)-\left(\begin{array}{c}2\\ 4\\ 8\end{array}\right)\;=\;\left(\begin{array}{c}-2\\ -13\\ -29\end{array}\right).\]2. On cherche les solutions de l'equation vectorielle \(\varphi(\mathbf{x})=\mathbf{0}\). En notant \(\mathbf{x}=(x,y,z)^{t}\), on obtient le systeme \[\left\{\begin{array}{rcl}0&=&-x+2y+2z,\\ 0&=&-8x+7y+4z,\\ 0&=&-13x+5y+8z.\end{array}\right.\] La seule solution de ce systeme est le vecteur nul, ce que l'on peut voir aussi en calculant le determinant de la matrice \[\left[\begin{array}{rrr}-1&2&2\\ -8&7&4\\ -13&5&8\end{array}\right].\] Donc \(\ker\varphi=\{\mathbf{0}\}\), l'unique base de \(\ker\varphi\) est \(\emptyset\), et sa dimension est nulle. +3. Puisque \(\ker\varphi=\{\mathbf{0}\}\), l'application \(\varphi\) est injective. Puisque les dimensions des espaces de depart et d'arriveee sont toutes deux egales a 3, \(\varphi\) est aussi surjective, et donc bijective. +4. En notant \(\mathbf{x}=(x,y)^{t}\), on a : \[(\varphi\circ\psi)(\mathbf{x}) = \varphi(\psi)(\mathbf{x}))\] \[= \varphi\left(\begin{array}{c}x-y\\ x+y\\ x+2y\end{array}\right)\] \[= \left(\begin{array}{c}-(x-y)+2(x+y)+2(x+2y)\\ -8(x-y)+7(x+y)+4(x+2y)\\ -13(x-y)+5(x+y)+8(x+2y)\end{array}\right)\] \[= \left(\begin{array}{c}3x+7y\\ 3x+23y\\ 34y\end{array}\right).\] + +**Solution de l'exercice 12** : + +1. Verifions que \(\varphi\) est lineaire : \[\varphi\left(\alpha\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\beta\left(\begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array}\right)\right)\] \[= \varphi\left(\begin{array}{c}\alpha x+\beta x^{\prime}\\ \alpha y+\beta y^{\prime}\\ \alpha z+\beta z^{\prime}\end{array}\right)\] \[= \left(\begin{array}{c}(\alpha y+\beta y^{\prime\prime})+( \alpha z+\beta z^{\prime})\\ \alpha x+\beta x^{\prime}\end{array}\right)\] \[= \alpha\left(\begin{array}{c}y+z\\ x\end{array}\right)+\beta\left(\begin{array}{c}y^{\prime}+z^{\prime}\\ x^{\prime}\end{array}\right)\] \[= \alpha\varphi\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\beta\varphi\left(\begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array}\right).\] Ensuite, \(\varphi(\mathbf{u})=(5,1)^{t}\), \(\varphi(\mathbf{v})=(2,1)^{t}\) et \[\varphi(\mathbf{u}-2\mathbf{v})=\left(\begin{array}{c}5\\ 1\end{array}\right)-2\left(\begin{array}{c}2\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ -1\end{array}\right).\]2. Le vecteur \((x,y,z)^{t}\) appartient a ker \(\varphi\) si et seulement si \(y+z=0\) et \(x=0\). C'est donc l'ensemble des vecteurs de la forme \((0,y,-y)^{t}\) ou \(y\in\mathbb{R}\) : \[\ker\varphi=\left\{\left(\begin{array}{c}0\\ y\\ -y\end{array}\right)\Bigg{|}\ y\in\mathbb{R}\right\}=\text{vect}\left\{\left( \begin{array}{c}0\\ 1\\ -1\end{array}\right)\right\}.\] Le sous-espace vectoriel ker \(\varphi\) est donc de dimension \(1\), et admet pour base le singleton \(\{(0,1,-1)^{t}\}\). +3. D'apres le theoreme du rang, \(\dim\mathbb{R}^{3}=\operatorname{rg}\varphi+\dim\ker\varphi\), ce qui implique que \(\operatorname{rg}\varphi=2\). On en deduit que \(\operatorname{im}\varphi=\mathbb{R}^{2}\) et que n'importe quelle base de \(\mathbb{R}^{2}\), par exemple la base canonique, est une base de \(\operatorname{im}\varphi\). + +**Solution de l'exercice 13** + +1. Si \(\mathcal{A}\) est liee, il existe \(\alpha_{1},\ldots,\alpha_{m}\) non tous nuls tels que \(\alpha_{m}x_{m}+\cdots+\alpha_{m}x_{m}=0\). Mais alors \[\alpha_{1}\varphi(x_{1})+\cdots+\alpha_{m}\varphi(x_{m})=\varphi(\alpha_{1}x_ {1}+\cdots+\alpha_{m}x_{m})=0,\] et puisqu'au moins un des \(\alpha_{j}\) est non nul, non voyons que \(\{\varphi(x_{1}),\ldots,\varphi(x_{m})\}\) est liee. +2. Ce point se deduit du precedent par contre-apposition. +3. Supposons \(\mathcal{A}\) libre et \(\varphi\) injective, et considerons l'equation \(\alpha_{1}\varphi(x_{1})+\cdots+\alpha_{m}\varphi(x_{m})=0\). Le membre de gauche n'est autre que \(\varphi(\alpha_{1}x_{1}+\cdots+\alpha_{m}x_{m})\), et puisque \(\varphi\) injective, on a necessairement \(\alpha_{1}x_{1}+\cdots+\alpha_{m}x_{m}=0\). Puisque \(\mathcal{A}\) est libre, on deduit de cette derniere equation que \(\alpha_{1}=\ldots=\alpha_{m}=0\). 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zZ^eZiufl(nqp&(&cxbKG8U!G~U&rH>T|N7xC{?~fD@JnEzN#2h+j!T3YFp1QqAt~R`t;%JcYk>Mf!dcZU;X>L z|Mxa3JidJO>-Xc`_FHepAh`G zAAkDOpZpAy?fB#CPtQO3@z2jc`aHz{`r=1F`r_lKuRnZ>!jkCQNQN)|?(hET-+unZ F{|63w_uK#g diff --git a/front_end/flagged/log.csv b/front_end/flagged/log.csv deleted file mode 100644 index af15e5a..0000000 --- a/front_end/flagged/log.csv +++ /dev/null @@ -1,9 +0,0 @@ -PDF File,output,flag,username,timestamp -flagged\PDF File\bc51931986aa4ff95e72\integration.pdf,"' - - ",,,2024-04-15 02:34:48.167890 diff --git a/front_end/main.py b/front_end/main.py index 481921e..03e432a 100644 --- a/front_end/main.py +++ b/front_end/main.py @@ -7,6 +7,7 @@ from langchain_community.embeddings.huggingface import HuggingFaceEmbeddings from backend.vector_db_manager import VectorDbManager from backend.inference import InferenceInstance +from backend.pdf_to_mmd import pdf_to_mmd import time @@ -22,8 +23,6 @@ port = get_accessible_port() # Launch a simple HTTP server to serve the PDF files - - def start_server(): command = ['python', '-m', 'http.server', f"{port}"] # Set the working directory to the documents folder to serve the PDF files @@ -57,12 +56,19 @@ def bot(history): global user_message_global, doc_path if doc_path != "": - print("FOUND DOC_PATH") - vector_db_manager.create_vector_store_from_pdf(doc_path) + print(f"FOUND DOC_PATH {doc_path}") + doc_extension = doc_path.split(".")[-1] + if doc_extension == "mmd": + vector_db_manager.create_vector_store_from_latex(Path(doc_path)) + elif doc_extension == "pdf": + vector_db_manager.create_vector_store_from_pdf(doc_path) + else: + print(f"Unsupported extension: {doc_extension}") else: print("NOT FOUND DOC_PATH") - bot_message = inference_instance.get_next_token(user_message_global, doc_path.split("\\")[-1]) + doc_name = Path(doc_path).stem + ".mmd" if math_checkbox.value else Path(doc_path).name + bot_message = inference_instance.get_next_token(user_message_global, doc_name) history[-1][1] = "" for message in bot_message: history[-1][1] = message @@ -70,11 +76,21 @@ def bot(history): yield history -def update_path(p): +def update_path(p, checked): """Update the global variable doc_path with the selected PDF path""" + print("Updating path") global doc_path - doc_path = str(p) - print(f"Selected PDF path: {doc_path}") + name = Path(p).name + stem = Path(p).stem + if checked: + if not (Path(r"../documents/mmds") / (stem + ".mmd")).exists(): + print(f"Converting {name} to MMD") + pdf_to_mmd(r"../documents/pdfs/" + name) + print(f"Selected DOC path: {stem}.mmd") + doc_path = r"../documents/mmds/" + stem + ".mmd" + else: + print(f"Selected DOC path: {name}") + doc_path = str(p) def pdf_viewer(pdf_file): @@ -107,6 +123,7 @@ with gr.Blocks() as main_tab: with gr.Row(): with gr.Column(scale=12): file_input = gr.File(label="Select a PDF file") + math_checkbox = gr.Checkbox(label="Enable math mode (your pdf file will be converted to some latex-like format for the chatbot to understand it better)") with gr.Column(): with gr.Group(): @@ -117,7 +134,7 @@ with gr.Blocks() as main_tab: ) file_input.change(pdf_viewer, inputs=file_input, outputs=pdf_output) - file_input.upload(update_path, inputs=file_input) + file_input.upload(update_path, inputs=[file_input, math_checkbox]) # Define options tab @@ -129,6 +146,21 @@ with gr.Blocks() as options_tab: gr.Textbox(label="Options", scale=2) -app = gr.TabbedInterface([main_tab, options_tab], ["Main", "Options"]) +# Define conversion tab +with gr.Blocks() as conversion_tab: + with gr.Column(): + file_input = gr.File(label="Select a PDF file to convert to MMD") + html_output = gr.HTML(label="Output") + + def upload_func(file_input): + name = Path(file_input).name + file_path = fr"../documents/pdfs/{name}" + pdf_to_mmd(file_path) + + + file_input.upload(upload_func, inputs=file_input) + + +app = gr.TabbedInterface([main_tab, options_tab, conversion_tab], ["Main", "Options", "Conversion"]) app.queue() app.launch()

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