Let U and V be subspaces of a vector space W . The sum of U and V , denoted U +

Question

Let U and V be subspaces of a vector space W . The sum of U and V , denoted U + V , is defined to be the set of all vectors of the form u + v, where u U and v V . Prove that U + V and U n V are subspaces of W . If U + V = W and then W is said to be the direct sum of U and V and we write W = U V . Show that every element w W can be written uniquely as w = u + v, where u G U and v V . Let U be a subspace of dimension k of a vector space W of dimension n. Prove that there exists a subspace V of dimension n - k such that W = U V . Is the subspace V unique? If U and V are arbitrary subspaces of a vector space W , show that dim(U + V ) = dim U + dim V - dim(U V ).

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http://xmlearning.maths.ed.ac.uk/lecture_notes/vector_spaces/vector_subspaces/vector_subspaces.php

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