Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is an element of V. Then show that 2(3,9) is not an element of V.

Q: Use the inner product (p, q) = aobo + azb1 + azb2 to find (p, a), ||P||, |||, and d(p, q) for the…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Use the inner product (p, a) = aobo + a,b, + azbz to find (p, 9), ||P||, ||9||, and d(p, q) for the…

A: a. -3 b. √6 c. √2 d. √11

Q: Need prove If R is a UFD and f(x) , g(x) are elements of R[x] , then c(fg) = c(f)c(g)

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Q: For every real m x n matriz mx m) and a diagonal m x n matriz

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Q: Find all numbers z such that Az3 + Bz² + Cz = D %3D B = [2 4 21 1 1 1. A = 1 3 1. 1 -2 2 4 21 C li 3…

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Q: Let P3 have the inner product given by evaluation at -7, -1, 1, and 7.

A: Given, ℙ3 have the inner product given by evaluation at -7, -1, 1 and 7. And P0(t)=3, p1(t)=2t,…

Q: Use the inner product = aobo+ aıb1+ azbz+ asb; to find , lIlp|], d(p, q), and then find the angle…

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Q: Describe all zero-divisors and units of Z ⨁ Q ⨁ Z.

A: Consider x = (a,0,0) for .  Similarly, if x has at least one zero component then we can find a…

Q: Find z such that: IZI = 2, arg Z : 6.

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Q: Find the value of the Conj uction g(+) ifya know that w(f.g)=3e" and fCo=et (Wronskain)

A: We will find out the required value of g(t).

Q: Does the following set of polynomials {t? + 1, t2 + t,t + 1} span P2?

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Q: In real analysis,Prove that if A is countable, then m(A) =0.

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Q: How did they calculate what T does to the elements of B?

A: You want to know "How did they calculate what T does to the elements of B?" We have  B=1, x, x2, x3…

Q: Use the inner product (p, q) = abo + a,b, + a,b, to find (p, q), ||p ||, ||9||, and d(p, q) for the…

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Q: For what value(s) of t does the equality (13 – 121, 0.333333f² + 1) = (0,5) hold true? t = (If there…

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Q: A−1 in the block form.

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Q: Prove: If f1, f2 ∈ R(a) on [a, b] with f1 ≤ f2, then ∫ab f1 da≤ ∫ab f2 da

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Q: Find (a) ⟨ p, q⟩, (b) ||p||, (c) ||q||, and (d) d( p, q) for the polynomials in P2 using the inner…

A: Given that, p(x) = 1 + x2, q(x) = 1 − x2 Compare the polynomial p(x) = 1 + x2 = 1 + 0x + x2 with…

Q: Show that is isomorphic to Z2.

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Q: Let * be the operation on the cartersian product of real numbers R *R such that (a,x) * (b,y) =…

A: Given that, let * be the operation on the cartesian product of real numbers R*R  such that (a, x) *…

Q: Find all (p, q, r) such that ((p + -4) evaluates to 1.

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Q: Suppose  v,w is an element of V. Explain why there exists a unique x which is an element of V such…

A: Suppose v, w is an element of V. Explain why there exists a unique x which is an element of V such…

Q: Find a pair of integers x and y such that 66x + 48y = gcd(66, 48)

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Q: Which ring is fx=2x^2-1 irreducible? Z[x] Q[x] Z2[x] Z3[x

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Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

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Q: Prove that P(A) - P(3)| 4 P(AAB). where A is the symmetric difference.

A: The following identity can be proved after breaking the LHS and RHS into mutually exclusive events,

Q: Find out the values of X1 and X3 from the following set of equations using Cramer's rule: 2X,…

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Q: Use the inner product (p, q) = aobo + a¡b1 + azb2 to find (p, q), ||p||, ||g||, and d(p, q) for the…

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Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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

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Q: Find all positive integers a, b such that a | b^2-b+1 and b | a^2-a+1

A: Solution :-

Q: Use the inner product (p, a) = aobo + ajb1 + azb2 to find (p, a), ||p||, ||9||, and d(p, q) for the…

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Q: Let P3 have the inner product

A: Given, pt=t2 and qt=2t-3. given by evaluation -2,-1,0,1, and 2. p-2=4q-2=-4-3=-7 p-1=1q-1=-2-3=-5…

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

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Q: Use the adjoint to compute A- for 2. 3 A = | 0 2 -1 1 -2

A: A=2310112-1-2 Now determinant value is, A=2-2+1-30-2+10-2=-2+6-2=2 Now cofactors are,…

Q: Use the inner product (p, q) = abo + a,b, + a,b, to find (p, q), ||p ||, ||9||, and d(p, q) for the…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts for…

Q: Use the inner product (o, 9) = 2obo +a1b1 +a2b2 to find o, 9), |P||- |||, and d(p, a) for the…

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Q: Use the inner product (p, q) = aobo + a¡b1 + azb2 to find (p, q), ||P||, ||9||, and d(p, q) for the…

A: The given polynomial is: p(x)=5+x2         and         q(x)=5-x2

Q: uppose A is a diagonali= . Would this be true if

A: Given, A is a diagonalizable n x n matrix whose eigenvalues are all 1 or -1.

Q: Prove that: ā x (B x č) = (ã-č )b – (ã ·5 )ẽ

A: See the details solution in below

Q: Prove that √2 is not an element of Q(?).

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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: Use the inner product (p, g) = aobo + a¡b1 + ażb2 to find (p, q), ||p||, ||9||, and d(p, q) for the…

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Q: Find A' for the given choice of U. If A={positive integers}, U=Z

A: We want to find  A' for the given choice of U. If A={positive integers}, U=Z

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: Q3) Use the inner product = a,bo + a,b̟ + a½b, to find , l\p||, d(p, q), and then find the angle…

A: According to question, 3. Inner product:p, q = a0b0 + a1b1 + a2b2px = x2 - x+1,qx = -3 x2 + x

Q: Q3) Use the inner product = a,bo + a,b1 + azb2 to find , llp||, d (p, q), and then find the angle…

A: Given: Inner product: p, q=a0b0+a1b1+a2b2 and the polynomials  p(x)=1-x+x2 and q(x)=x-3x2. To find:…

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

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Q: Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p||, ||q||, and d(p, q) for the…

A: We have to solve given Questions.