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LA06Homework5

# LA06Homework5 - Practice Problems(1 Let u and v be two...

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Practice Problems 2/13/06 (1) Let u and v be two distinct vectors of a vector space V . Let { u, v } be a basis for V and a, b nonzero scalars. Show that { u + v, au } and { au, bv } are also bases for V . (2) The set of all (3 × 3) matrices having trace equal to zero is a subspace W of M 3 × 3 . Find a basis for W . Solution: Note that any (3 × 3) matrix a b c , can be written as a linear combination of 0 1 0 0 0 0 0 0 0 , 0 0 1 0 0 0 0 0 0 , 0 0 0 1 0 0 0 0 0 , 0 0 0 0 0 1 0 0 0 , 0 0 0 0 0 0 1 0 0 , 0 0 0 0 0 0 0 1 0 , 1 0 0 0 - 1 0 0 0 0 , 0 0 0 0 - 1 0 0 0 1 . This is a linearly independent, generating set. Hence the dimension of W is 8. (3) The set of all (3 × 3) upper triangular matrices is a subspace W of M 3 × 3 . Find a basis for W . (4) Let V be a vector space having dimension n , and let S be a subset of V that generates V . Prove that there is a subset of S that is a basis for V . (Do not assume S to be finite.) (5) Let W 1 , W 2 be subspaces of a finite dimensional vector space V .

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LA06Homework5 - Practice Problems(1 Let u and v be two...

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