Find a bosis for the nullspace of the morkit 5-23 (-12 -2-10 4 -6J

Q: Consider the following. 13 -60 -4 -5 A = P = 3 -14 (a) Verify that A is diagonalizable by computing…

A: Given:    A=13-603-14    P=-4-5-1-1   P-1=-4-5-1-1-1

Q: Find (a) ATA and (b) AAT. Show that each of these products is symmetric.

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Q: 1 0 Find a mat rixP such that P- 0 2 | 0 2

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Q: Let A = AB= 1 -1 2 0 5 1 -2 0 −1 14 -57 3 [-2 27 -32 0 -5 2 -1 1 0 -7 1 X -2 and B = 5 2 Find x such…

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Q: Find bases for the null space and row space of A. 5. 8. -7 -4 -1 -8 1 3. 3 A = -1 0. -1 1 -2 4 7. 10…

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Q: know there exists at least one c in the open values of c that work.

A: We have to find the following conditions according to the given information

Q: Prove by contradiction: “V3 is iirational." « 3

A: In this, we need to prove that 33 is a irrational number. So, we need to assume that 33 is rational…

Q: 7. Let B₁, be the set of bitstrings of length n. Describe a bijection f: P({1,2,...,n}) → B₁.

A: The idea is to look at the indicator function of a given set. The detailed solution is presented as…

Q: Show that is isomorphic to Z2.

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Q: is bijection. Do the same thing with Z,.

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Q: Prove that x4 + 15x3 + 7 is irreducible over Q

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Q: (b) Show that 1+ 2/2 has infinite order in R*.

A: Given that R=m+n2 | m,n∈ℤ (b) The objective is to show that 1+22 has infinite order in ℝ×

Q: Show that (AB)¹ = BTA¹ if A = 2 0 2 0 and B = -1 1 15) -3 1 -1 -1 2 2). a

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Q: Verify that there is no magic square of order 2.

A: We will verify that there is no magic square of order 2 using the proof by contradiction.

Q: Show that 2Z ∪ 3Z is not a subring of Z.

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: FOR ANY TWOCOMPLEX NUMBERSZ, ANDZ,,PROVE THAT |Z|+|Z,|=,(Z, +Z,)+ /Z,Z» +÷(Z, +Z»)- Z,Z, .

A: for two complex number Z1 and Z2 prove that Z1+Z2=12Z1+Z2+Z1Z2+12Z1+Z2-Z1Z2

Q: Give an example of a nonzero 2x2 matix A such that A2=O

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Q: Determine which of the four inner product axioms do not hold. Give a specific example in each case.

A: Given that, u=u1u2 and v=v1v2 in R2 Giving inner product u·v=u1v1-u2v2 Axiom (4) of inner product…

Q: Find bases for the null and range spaces of 1 2 3 1 1 1 A = 3 4 5

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Q: If A is diagonalizable, then A is similar to D. Select one: O True O False

A: Given   if matrix a is diagonalizable then A is SIMILAR  to D

Q: Find gcd (272, 1479) and find the integers x, y such that gcd(272, 1479) = 272x + 1479y

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Show that U(8) is isomorphic to U(12).

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Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: Determine A-" if A is invertible. 1 0 -2 4 -2 1 1 2 -10

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Q: prove that 2,

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Q: Find (a) ATA and (b) AAT. Show that each of these products is symmetric.

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Q: Show that A° = A°-0- and A-o = A- %3D

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Q: Write all essential submodule of Z 24 as Z-module.

A: (C) We know that a submodule E of M is essential if it has a non trivial intersection with every non…

Q: Show that  ℝ2 = span([1 1], [1 -1])

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Q: is the columd of matnla A Explain why Oineanly nメn an A is invertible. independent when are 1 6

A: To explain why the columns of an n x n matrix A are linearly independent when A is invertible.

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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Q: is in the null space of P Yo T yo is in the null space of P O yo is in the null space of P

A: Given that P is linear operator

Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

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Q: Prove that =0

A: ∑i=0ne-λλxx!=1n!∫λ∞ e-xxn    ...............(1) We will proof this by using induction on n. Here,…

Q: Find (a) ATA and (b) AAT. Show that each of these products is symmetric.

A: Given, A matrix is,

Q: Find the all 7 principal minors (PMs) of (I- A)

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Q: Prove that 0<1.

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Q: Prove the

A: Follow the steps that I listed given below pitcher.

Q: Produce an example of a subset of 1R³ that is Closed under addition, but not closed under scalar…

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Q: Show that If |z| = 1, then: -1 = Z

A: z is a complex number. z=a+ib |z|=(a+ib)(a-ib)     =a2-b2 if |z|=1, then a2+b2=1

Q: Show that for A and B of n×n order with complex enteries the matric Ab need not be similar to BA ,…

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Q: Construct two bijections from (0,1) to (1,1/0).

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Q: Theorem 4P.1 There are exactly six lines in the four-point geometry.   proof the Theorem 4P.1 by…

A: To prove that in the Four point geometry , there are exactly six lines: the proof should be based on…

Q: Find bases for the null space and row space of A. 3 4 6. -10 -8 -4 -16 A = -1 -1 0 -2 3 4 7 13 14

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Q: 2. Determine the greatest common divisor of a(x) = x³ – 2 and b(x) = x + 1 in Q[x] and write it as a…

A: Given the polynomials   ax=x3-2 and bx=x+1 to find gcd(a(x), b(x)) and write it as a linear…

Q: Find the minimal polynomial satisfied by √2 + √5. over Q.

A: We have to find the minimal polynomial which satisfied by 2+5 over Q.

Q: let FQ) = -2Xidefined on [14], then UC Pmif) =

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.