By E. R Kolchin
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Let V = R2 . If Exercises 33 Exercise 69 Let V = R and let ◦ be an operation on R defined by a ◦ b = a 3 b. Is V , together with the usual addition and “scalar multiplication” given by ◦, a vector space over R? Exercise 70 Show that Z is not a vector space over any field. Exercise 71 Let V be a vector space over the field GF(2). Show that v = −v for all v ∈ V . Exercise 72 In the definition of a vector space, show that the commutativity of vector addition is a consequence of the other conditions.
3) Recursively, calculate f0 (X)g0 (X), f1 (X)g1 (X), and (f0 + f1 )(X)(g0 + g1 )(X). (4) Then f (X)g(X) = X n (f1 g1 )(X) + X n/2 (f0 + f1 )(g0 + g1 ) − f0 g0 − f1 g1 (X) + (f0 g0 )(X). 59 arithmetic operations. If n is sufficiently large, the difference between these two bounds can be significant. 46 4 Algebras Over a Field The main idea of Karatsuba’s algorithm lies in the recursive reduction of the degrees of the polynomials involved. The method of recursive reduction has since been extended to fast algorithms in many other areas of mathematics.
Is V a vector space over R? Exercise 63 Let V = {i ∈ Z | 0 ≤ i < 2n } for some given positive integer n. Define operations of vector addition and scalar multiplication on V in such a way as to turn it into a vector space over the field GF(2). Exercise 64 Let V be a vector space over a field F . Define a function from GF(3) × V to V by setting (0, v) → 0V , (1, v) → v, and (2, v) → −v for all v ∈ V . Does this function, together with the vector addition in V , define on V the structure of a vector space over GF(3)?