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MATH 330H 41

# MATH 330H 41 - Math 330 Homework 4.1(Pages 223,224 For...

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Math 330 Homework 4.1 (Pages 223,224) For problems (6) and (8) decide if the set is a subspace of P n for an appropriate value of n . Justify your answer. (6) All polynomials of the form p ( t ) = a + t 2 where a R . SOLUTION: This is not a vector space since the sum of two polynomials of this form is never of this form. For example, although 1 + t 2 belongs to this collection, (1 + t 2 ) + (1 + t 2 ) = 2 + 2 t 2 does not. (8) All polynomials in P n such that p (0) = 0 . SOLUTION: This is a vector space. Suppose that p and q are both polynomials and p (0) = q (0) = 0. Then ( p + q )(0) = p (0) + q (0) = 0 + 0 = 0. Suppose further, that λ is any real number. Then ( λp )(0) = λ ( p (0)) = λ · 0 = 0. Therefore, scalar multiplication and addition preserves the criteria to be in this collection so this collection is indeed a vector space. In problems 16 and 18, let W be the set of all vectors of the form shown where a, b , and c represent arbitrary real numbers. Either find a set of vectors which spans W or explain why W is not a vector space. (16) - a + 1 a - 6 b 2 b + a SOLUTION: This is not a vector space. Notice that although the following two matrices separately satisfy

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