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HWSolutions4.1

# HWSolutions4.1 - Newberger Math 247 Spring 03 Homework...

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Newberger Math 247 Spring 03 Homework solutions: Section 4.1 #5-8 In Exercises 5-8 determine if the given set is a subspace of P n for an appropriate value of n . Justify your answer. 5. All polynomials of the form p ( t ) = at 2 . The set of all polynomials of the form p ( t ) = at 2 is all scalar multi- ples of the polynomial t 2 , thus this set is just Span { t 2 } . Since all spans are subspaces, this set is a subspace. 6. All polynomials of the form p ( t ) = a + t 2 . Let H denote the set of all polynomials of the form p ( t ) = a + t 2 . Then we can check that the zero polynomial 0 + 0 t + 0 t 2 is not in H . We try to find a number a such that a + t 2 = 0 + 0 t + 0 t 2 . Since the coefficient on t 2 on the LHS is 1, but the coefficient on t 2 on the RHS is 0, these can never be equal (since 0 6 = 1 ). Thus the zero polynomial is not in H , since there is no value of a that will make a + t 2 = 0+0 t +0 t 2 . Since H does not contain the zero polynomial, H is not a subspace.

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HWSolutions4.1 - Newberger Math 247 Spring 03 Homework...

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