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Unformatted text preview: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 1 Posted: Fri. Oct. 12, 2007 Written by: Luca Candelori Exercise 1. (a) TRUE. This is one of the vector space axioms (VS 3). (b) FALSE. We proved this in class but here is a short proof. Suppose there is another 0 vector, call it 0 . Then for any vector x , x + 0 = x x + 0 = x and therefore x + 0 = x + 0 which, by the Cancellation Law, implies that 0 = 0 . (c) FALSE. If x is the zero vector, this certainly does not hold. (d) FALSE. Again, it does not hold if a = 0. (e) TRUE. An element in F n consists of n coordinates t 1 ,...,t n taken from F . For example 1 2 can be regarded as an element of R 3 . But this is also a matrix with 3 rows and 1 column so, in general, an element of F n can be regarded as a matrix with n rows and 1 column. This is called the columnn vector representation. You can also represent it as a matrix with n columns and 1 row, in which case we call it a row vector, but this is less common. In fact, we will see later that row vectors are used to represent elements of a the dual space, a vector space that stems quite naturally from any other vector space. (f) FALSE. An m n matrix has m rows and n columns. Dont get confused! (g) FALSE. If p has degree d and q has degee e , then p + q is still a polynomial, so it is a member of P ( F ). (h) FALSE. If p has degree d and q has degee e , then p + q has degree max( d,e ), without equality. For example, in P ( R ),- x 2 +3 and x 2- x + 7 are both degree 2 polynomials, but when added they give a polynomial of degree 1. (i) TRUE. Write p ( x ) = a n x n + ... + a , where a n is nonzero so that p has degree n . Then c p ( x ) = ( ca n ) x n + ... + ca is also of degree n , since c,a n 6 = 0 implies that ca n 6 = 0 1 (j) TRUE. A polynomial of degree 0 in P ( F ) has the form a , for a F . Therefore one element of F determines a polynomial of degree 0 in P ( F ) completely....
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