Algorithmic Mathematics

Exercise 6.1. Let F = Z/(2). (a) Write down all the elements of F3. (b) Write down all the sequences in echelon form of elements of F2. Exercise 6.2. Let wi = (1,1,0), W2 = (2,2,2) be vectors in (Z/(3))", and let W (W1, W2) (a) Write down all the elements of W. (6) For each we W, determine ldindx(w), and if w #0, determine ldterm(w). Exercise 6.3. Let u, w1, W2, W3 Q3, with v = (4/7,-3/5,5/7); w1 = (1,1/2,1), W2 = (0,1,2), W3 = (0,0,1). (a) Trace Sift(v, (W1, W2, W3)). (b) Write v as a linear combination of w1, W2, W3, verifying your calculation explicitly. Exercise 6.4. Let F =Z/(11), and let W1 = (1,0, 7,3) W2 = (0,1,3,4) W3 = (0,0,0,1) EF4 (a) Use the algorithm Sift to prove that (5,2,4,3) & (W1, W2, W3). (b) Use the algorithm Sift to prove that v = (6,2,4,3) E (W1, W2, W3). Hence write v as a linear combination of w1, W2, W3, verifying your calculation explicitly. Exercise 6.1. Let F = Z/(2). (a) Write down all the elements of F3. (b) Write down all the sequences in echelon form of elements of F2. Exercise 6.2. Let wi = (1,1,0), W2 = (2,2,2) be vectors in (Z/(3))", and let W (W1, W2) (a) Write down all the elements of W. (6) For each we W, determine ldindx(w), and if w #0, determine ldterm(w). Exercise 6.3. Let u, w1, W2, W3 Q3, with v = (4/7,-3/5,5/7); w1 = (1,1/2,1), W2 = (0,1,2), W3 = (0,0,1). (a) Trace Sift(v, (W1, W2, W3)). (b) Write v as a linear combination of w1, W2, W3, verifying your calculation explicitly. Exercise 6.4. Let F =Z/(11), and let W1 = (1,0, 7,3) W2 = (0,1,3,4) W3 = (0,0,0,1) EF4 (a) Use the algorithm Sift to prove that (5,2,4,3) & (W1, W2, W3). (b) Use the algorithm Sift to prove that v = (6,2,4,3) E (W1, W2, W3). Hence write v as a linear combination of w1, W2, W3, verifying your calculation explicitly