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ass4 - MATH 223 Linear Algebra Fall 2007 Assignment 4...

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MATH 223, Linear Algebra Fall, 2007 Assignment 4 Solutions 1. Consider the vector space V = P 5 ( R ) of polynomials with real coefficients (in one variable t ) of degree at most 5 (including the zero polynomial). Show that if c ∈ R is any real number, then the set { 1 , t - c, ( t - c ) 2 , ( t - c ) 3 , ( t - c ) 4 , ( t - c ) 5 } is a basis of V . Solution: Since the set { 1 , t, t 2 , t 3 , t 4 , t 5 } is visibly a basis of V , we know that this vector space has dimension 6. To show that the 6 vectors { 1 , t - c, ( t - c ) 2 , ( t - c ) 3 , ( t - c ) 4 , ( t - c ) 5 } are a basis of V , it therefore suffices to show that they are linearly independent (as spanning is then automatic). So suppose that for some scalars a 0 , a 1 , a 2 , a 3 , a 4 , a 5 we have a 0 + a 1 ( t - c ) + a 2 ( t - c ) 2 + a 3 ( t - c ) 3 + a 4 ( t - c ) 4 + a 5 ( t - c ) 5 = 0; we wish to show that each a i must be zero. Comparing coefficients of t 5 on both sides, we see that we must have a 5 = 0. Comparing coefficients of t 4 on both sides of the resulting equation then forces a 4 = 0 as well. Continuing in this manner, we find that a 3 = a 2 = a 1 = a 0 = 0, as desired. 2. Let V be as in problem (1). Let W be the subset of V consisting of those polynomials p ( t ) such that p (0) = 0, and let U be the subset of polynomials p ( t ) that are even , i.e. such that p ( t ) = p ( - t ). (a) Show that U and W are subspaces of V . (b) Compute dim V , dim W , dim U , dim U W . Solution: (a) It suffices to show that each set U, W contains the zero vector and is closed under addition and scalar multiplication. Since the value of the zero polynomial at zero is indeed zero, W contains zero; similarly, as 0 = - 0, we see that U contains zero as well. Now suppose that
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