Let V be a vector space, and let U CV be a linear subspace. Define the...

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Let V be a vector space, and let U CV be a linear subspace. Define the following relation on V: x ~ y if and only if x – y E U 1. Prove this is an equivalence relation. The quotient of V by this equivalence relation is denoted by V/U, and there is a natural quotient map T : V – V/U, defined by v + [v]. 2. Define a vector space structure on the set V/U such that the quotient map is a linear map. Now let V be a finite dimensional vector space, and let A :V W be a linear map. 3. Prove that V/U is finite dimensional and that dim(V/U) = dim V – dim U. 4. Prove that if U'CV is such that V = U eU', then T defines an isomorphism from U' to V/U. 5. Prove that such a U' must exist. Let V be a vector space, and let U CV be a linear subspace. Define the following relation on V: x ~ y if and only if x – y E U 1. Prove this is an equivalence relation. The quotient of V by this equivalence relation is denoted by V/U, and there is a natural quotient map T : V – V/U, defined by v + [v]. 2. Define a vector space structure on the set V/U such that the quotient map is a linear map. Now let V be a finite dimensional vector space, and let A :V W be a linear map. 3. Prove that V/U is finite dimensional and that dim(V/U) = dim V – dim U. 4. Prove that if U'CV is such that V = U eU', then T defines an isomorphism from U' to V/U. 5. Prove that such a U' must exist.