Question 0.3. Compute the determinant of the matrix A = 0 1 2 0 3 1 in F5. Does this indicate if the matrix A is invertible or not? Next, give bases for the kernel and image of A over F5. You should find that the image is two dimensional and the kernel is one dimensional. We now discuss error correcting codes. Alice and Bob anticipate that they are going to want to communicate through an unreliable method where some part of the message may be garbled. Precisely, Alice expects that she will want to send one of N possible messages, which she will do using n characters from an alphabet of size p. She is concerned that as many as d - 1 of the characters may be received wrong. She will therefore try to choose N of the possible pr strings, in such a way that no string can be turned into any other by changing fewer than d characters. That is, any d 1 errors should be detectable. For example, take N = 8, p = 2, n = 7. Alice could choose the following 8 codewords to represent the messages she might send: Message Code word 0000000 Hello. 1001101 0101011 1100110 Goodbye. Safe! See you tomorrow. 0010111 It will rain tomorrow. 1011010 The final exam is easy. 0111100 1110001 I will leave soon. Help! Consider the linear space Fr, where the elements are (x1, x2,...,xn) where each x; Є Fp. (It is analogous to Rn.)

Q: Find Im(z) if z(8 + i) = 7+ (384 + 448i)

A:

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: 4444 is divided by a 1.19) find the remainder when 4444 * 2?=-1 (mod a).

A: 1.19 We have 4444  ≡  7 (mod 9). So, 44444444 ≡ 7 4444 (mod 9) Now,       73 ≡1 (mod 9) Or,…

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A: Here draw pascal triangle for some 1 to 4 rows and conclude for nth row

Q: 1. Find irr(a, Q) and deg(a, Q) for the given algebraic number a = √2+ i C. [Hint: Find irr(a, Q) as…

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Q: QUESTION 9 Use Mathematical induction to prove that 2 divides n2+5n, n21.

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Q: Compute ||w| using w= ||w|| || = 7 5 -4 (Type an exact answer, using radicals as needed.) LO

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Q: 1 Use the factorization A = PDP' to compute A^, wherek represents an arbitrary integer. а 11(b - а)…

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Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: (c) Suppose n EN is divisible by 5. What is (25n) (n) L?

A: Let, the prime factorization of n be as below.          n = p1a1 p2a2 . . . . . . pkak     for some…

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A: We have to prove that one of the integers a,a+2,a+4 is divisible by 3.

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Q: Prove (without using induction) F1- F = Fn-1Fn+2•

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Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

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A: Logarithm simplification

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Q: 1) Name the polynomial a) 3x? + 9x - 1 b) 2x* - 9x? - 1

A: a) 3x2+9x-1The above polynomial has a degree 2⇒It is quadratic polynomial