By Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin
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Extra info for Algebra and number theory : an integrated approach
Corollary. Let a, b be integers, let d = GCD(a, b) and let a= da1, b = dh. Then a1 and h are relatively prime. Proof. If GCD(a1, b1) = c > 1 then a1 = caz and b1 = cbz, where h, bz E Z. Then de> d is a common divisor of a and b, contrary to the definition of d. The result follows. Next we establish some further facts about relatively prime integers. 9. Corollary. Let a, b, c be integers. (i) If a divides be and a, b are relatively prime, then a divides c. (ii) If a, b are relatively prime and a, c are also relatively prime, then a and be are relatively prime.
4 we finally see that rk divides a and b. Now let u be an arbitrary common divisor of a and b. The equation r 1 = a - bq 1 shows that u divides r 1. The next equation r2 = b - r1 q2 shows that u divides r2. 4 we finally see that u divides rk. This means that rk is a greatest common divisor of a and b. Again this claim can be proved more formally using the Principle of Mathematical Induction. 6 proves that there are integers x, y such that rk = ax + by. It is important to note that the Euclidian Algorithm allows us to find these numbers x andy.
3. Find all matrices A E M2 (IR) with the property that A 2 = 0. 4. If we interchange rows j and k of a matrix A, what changes does this imply in the matrix AB? 5. If we interchange columns j and k of a matrix A, what changes does this imply in the matrix AB? 6. If we add a times row k to row j in the matrix A, what changes does this imply in the matrix AB? 7. 8. 9. 11. 10. 12. 13. Prove that for any square matrix A the product AA 1 is a symmetric matrix.
Algebra and number theory : an integrated approach by Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin