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If, furthermore, X is metrizable, then so is X/ M. Then X/ M is a locally convex space, and the topology on it is the quotient topology.
#CODIMENSION OF A SUBSPACE DEFINITION MOD#
The space obtained is called a quotient space and is denoted V / N (read V mod N or V by N ). In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by 'collapsing' N to zero. If W is a subspace of a finite-dimensional vector space V, then the codimension of the subspace W in the space V is the difference dimV - dimW between the. The mapping that associates to v ∈ V the equivalence class is known as the quotient map.Īlternatively phrased, the quotient space \displaystyle Short description: Vector space consisting of affine Subspaces. These operations turn the quotient space V/ N into a vector space over K with N being the zero class. do not depend on the choice of representatives).
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It is not hard to check that these operations are well-defined (i.e. Scalar multiplication and addition are defined on the equivalence classes by The quotient space V/ N is then defined as V/~, the set of all equivalence classes over V by ~. The equivalence class – or, in this case, the coset – of x is often denoted From this definition, one can deduce that any element of N is related to the zero vector more precisely, all the vectors in N get mapped into the equivalence class of the zero vector. That is, x is related to y if one can be obtained from the other by adding an element of N. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. Received JIf AZ is a subspace of a tensor product of vector spaces, A 0 B, we define of the rank r(M) inf rank X(X E M - (0)). Let V be a vector space over a field K, and let N be a subspace of V.