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Unformatted text preview: L . 3. [5 marks] A hyperplane in R n is a subspace of the form x 1 . . . x n : a 1 x 1 + + a n x n = 0 , where the scalars a 1 ,...,a n are not all zero. Prove that such hyperplanes are of dimension n-1, and that every subspace in R n of dimension n-1 is equal to a hyperplane. 4. [5 marks] Let V be a vector space, and U 1 ,U 2 V be subspaces. Prove that dim( U 1 + U 2 ) = dim U 1 + dim U 2-dim( U 1 U 2 ) . (You need not prove that U 1 + U 2 and U 1 U 2 are subspaces; your proof from question D3 of Section 3-4 is easily generalized.)...
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- Spring '09