hmw1 - Solutions to Homework 1. Math 110, Fall 2006. Prob...

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Solutions to Homework 1. Math 110, Fall 2006. Prob 1.2.1. (a) True – this is axiom (VS 3). (b) False – we proved the zero vector is unique. (c) False: take x to be the zero vector, and take any scalars a and b . (d) False: take a to be the zero scalar and any vectors x and y . (e) True, a column vector is also a matrix with 1 column. (f) False: it has m rows and n columns. (g) False, polynomials of different degrees can be added as in p.10. (h) False, the leading terms may cancel each other. (i) True, the leading term is multiplied by a nonzero scalar c . (j) True: the scalar c is identified with the constant function f ( x ) = c for all x . (k) True – that’s how functions are defined. Prob 1.2.8. By two laws (VS 1) and (VS 8) of distributivity, we have ( a + b )( x + y ) VS 7 = a ( x + y ) + b ( x + y ) VS 8 = ax + ay + bx + by. Prob 1.2.10. First observe that the sum of two differentiable functions is differentiable, and the product of a real number and a differentiable function is also differentiable. Hence V is closed under addition and multiplication by scalars. Axioms (VS 1), (VS 2), (VS 5), (VS 6), (VS 7), (VS 8) are satisfied by functions,
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hmw1 - Solutions to Homework 1. Math 110, Fall 2006. Prob...

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