Quiz — Sous-espaces vectoriels de dimension finie (5 exercices).pdf

Question 1 sur 10 20:00

[{"id":86143,"question":"Soit E un sous-espace vectoriel de dimension 3 dans ℝ⁵. Quelle est la dimension maximale possible de F, un sous-espace de E de dimension 2 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dimension d'un sous-espace F de E ne peut pas dépasser celle de E. Ici, dim(F) ≤ dim(E) = 3, donc la dimension maximale de F est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":86144,"question":"Si deux sous-espaces E et F de ℝⁿ vérifient dim(E) + dim(F) \u003E n, alors E ∩ F est nécessairement non réduit à {0}.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai d'après le théorème de la dimension : dim(E + F) = dim(E) + dim(F) - dim(E ∩ F) ≤ n, donc dim(E ∩ F) ≥ dim(E) + dim(F) - n \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":86145,"question":"Soit F = {(x, y, z) ∈ ℝ³ | x + y + z = 0}. Quelle est la dimension de F ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"F est un plan vectoriel (sous-espace de dimension 2) dans ℝ³, car il est défini par une seule équation linéaire non triviale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":86146,"question":"La somme de deux sous-espaces vectoriels E et F est toujours un sous-espace vectoriel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la somme E + F = {u + v | u ∈ E, v ∈ F} est toujours un sous-espace vectoriel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":86147,"question":"Soit E un sous-espace vectoriel de dimension 4 dans ℝ⁶. Quelle est la dimension minimale possible de F, un sous-espace de E de dimension 3 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dimension minimale de F est 3, car F est un sous-espace de E de dimension 3. Il n'y a pas de contrainte supplémentaire ici.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":86148,"question":"Si E et F sont deux sous-espaces vectoriels de ℝⁿ tels que E ∩ F = {0}, alors dim(E + F) = dim(E) + dim(F).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : c'est une conséquence directe de la formule de Grassmann lorsque E ∩ F = {0}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":86149,"question":"Soit E = vect{(1, 0, 1), (0, 1, 1)} et F = vect{(1, 1, 0), (1, -1, 0)} dans ℝ³. Quelle est la dimension de E + F ?","option_a":"2","option_b":"3","option_c":"4","option_d":"Impossible à déterminer","option_e":"","option_f":"","bonne_reponse":"b","explication":"E + F = ℝ³ car les vecteurs de E et F engendrent tout ℝ³. La dimension est donc 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"Impossible à déterminer\"}}","_debug_options_count":4},{"id":86150,"question":"Un sous-espace vectoriel de dimension finie peut avoir une infinité de bases différentes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : une base n'est pas unique, mais toutes les bases d'un même sous-espace ont le même nombre d'éléments (la dimension).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":86151,"question":"Soit E un sous-espace vectoriel de dimension 2 dans ℝ⁴. Quelle est la dimension maximale possible de F, un sous-espace de E de dimension 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dimension maximale de F est 1, car F est un sous-espace de E de dimension 1. Il ne peut pas avoir une dimension supérieure.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":86152,"question":"Si dim(E) = dim(F) = 3 dans ℝ⁵, alors dim(E ∩ F) peut être égal à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : par exemple, si E et F sont deux plans distincts dans ℝ⁵, leur intersection peut être une droite (dimension 1) ou un point (dimension 0), mais aussi un plan (dimension 2) si E = F.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]