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Unformatted text preview: Math110 Prof. Grunbaum GSI: E. Wayman(DIS107) Homework 1 Kyung Mo Kweon January 26th, 2011 1.2 Vector Spaces 1 . Label the following statements. (a) Every vector space contains a zero vector. Answer: True. Trivial. A zero vector meets all of the conditions (VS1) through (VS8) on page 7. 0 = 0 , 0 + 0 = 0, x 0 = 0 where x V (b) A vector space may have more than one zero vector. Answer: False. Assume there exists two zero vectors, say 0 and 0 . First by (VS1), we know there exists a vector x such that 0 + x = x +0. Then, there exists a vector y such that x + y = 0 by (VS4). Then, 0 = 0 + 0 = 0 + ( x + y ) = (0 + x ) + y = (0 + x ) + y = 0 + ( x + y ) = 0 + 0 = 0 . Consequently, 0 = 0. Thus, zero vector is unique. (c) In any vector space, ax = bx implies that a = b . Answer: False. Suppose x = 0. Then even if a 6 = b , it still qualifies ax = bx . (d) In any vector space, ax = ay implies that x = y . Answer: False. Same as above. Suppose a = 0. (e) A vector in F n may be regarded as a matrix in M n 1 ( F ). Answer: True. According to example 1, vectors in F n may be written as column vectors which are n 1 matrices. (f) An m n matrix has m columns and n rows. Answer: False. m rows and n columns. (g) In P ( F ), only polynomials of the same degree may be added. Answer: False. f ( x ) = ax 2 + bx + c and g ( x ) = d . Both have degree of 2 and 0 respectively, but it can be added together. 1 Math110 Prof. Grunbaum GSI: E. Wayman(DIS107) (h) If f and g are polynomials of degree n , then f + g is a polynomial of degree n . Answer: False. Counterexample: f ( x ) =- ax + b and g ( x ) = ax + b . the degree of both is 1. Yet, f ( x ) + g ( x ) = 2 b of which the degree is zero. (i) If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n . Answer: True. f ( x ) = a n x n + a n- 1 x n- 1 + + a = cf ( x ) = ca n x n + ca n- 1 x n- 1 + + ca = b n x n + b n- 1 b n- 1 + + b . (j) A nonzero scalar of F may be considered to be a polynomial in P( F ) having degree zero. Answer: True. A nonzero scalar of F is f ( x ) = c for some c . Since the largest exponent of x is 0, the degree is zero. (k) Two functions in F ( S,F ) are equal iff they have the same value at each element of S . Answer: True. Let two functions are f and g . They have the same value at each s S = f ( s ) = g ( s ). By example 3, they are equal. 16 . Let V denote the set of all m n matrices with real entries; so V is a vector space over R by example 2. Let F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix addition and scalar multiplication? Answer: Yes. Even if F is the field of rational numbers, it still qualifies that for 1 i m and 1 j n , ( A + B ) ij = A ij + B ij and ( cA ) ij = cA ij where A,B M m n ( F ) and c F . It is also trivial because Q R . Therefore, the same example used in example 2 can be used again because those numbers are also rational.example used in example 2 can be used again because those numbers are also rational....
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