2. Determine if the function (u, v) = µ₁ V₂ + U2V1, where ủ = (u₁, U₂) and ở = (V₁, V₂) defines an inner product on R2². Check each axiom to determine if it holds or fails. Show all work.

Q: assume that Pr[A′]=0.55, Pr[B]=0.35 , and Pr[A′∩B]=0.2. (1) Pr[A∩B]=Pr[A∩B]=   (2)…

A: Solution : Given information; P(A')=0.55 P(A)=1-P(A')=0.45P(B)=0.35 and P(A'∩B)=0.2

Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

A:

Q: 8. The polynomials x - 1, (x - 1)2, and (x - 1)³ span R3[r]. 9. If S and T are finite spanning sets…

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Q: Show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

A: We have to show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v…

Q: а. v(Ã ·) = Ã × (V x B) + B x (v x Ä) + (Ã ·v)Ē + (B · V)Ā b. · (Ã × B) = B · (v × Ã) – Ã · (V × B)…

A: Part(a): ∇A⇀·B⇀=A⇀×∇×B⇀+B⇀×∇×A⇀+A⇀·∇B⇀+B⇀·∇A⇀ From the left-hand side of the given equation,…

Q: If (0,1) ~ (-1,1) and (-1,1) ~ R. Deduce that (0,1) ~R

A:

Q: Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...)…

A: Here, in the question, there are given the bounded linear operators l1→l∞ which are given as…

Q: Let T : R? → R² such that T (1,0) = (3, 1), and T (0, 1) = (-1,3). Find T (2, 5). T (2, 5)=( (enter…

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Q: Let A ∈ M (3,R) be invertible and C, D ∈ M (3,2,R) . Solving B from the equality A(B+Ct)t = D, what…

A:

Q: Let A ∈ Rm×n, B ∈ Rn×r, and x, y ∈ Rn. Supposethat the product AxyTB is computed in the…

A: Let's first calculate the number of scalar additions and multiplications required for each method:…

Q: 1. Let Q(v2) = {a+b/2| a, b E Q}. Prove that Q(/2) = Q[x]/(x² – 2).

A: We have to show that isomorphism means there is function ϕ such that we have to show that it is an…

Q: Which elements are invertible in Z4[x,y]?

A: W need to find elements which are invertible inZ4x,y.

Q: Let j = 7 and U=span(ü), with ü = -6. Write j = ã1+ã2, where 1 in U and az in U-.

A:

Q: Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) =…

A: Let P3 have the inner product given by evaluation at -2, -1, 0, 1 and 2 . Let p(t) = t2-2t  and q(t)…

Q: Determine the kernel and the range of the linear operator on L(x) = (x1 + x3, X2, 0)

A:

Q: Q.4.(a). Show that the following functions is not an inner product on R3, where u = (u, U2, U3) and…

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Q: x · y = (x + y² – x² - y²)

A:

Q: 3. Write the cov(a + bx, c + dY) in terms of var(X), var(Y), and cov(Y,X) (Math proof)

A: Using the properties of covariance, the expression simplifies to:cov(a + bX, c + dY) = ac * cov(X,…

Q: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: Find the degree of Q(V2, V5) over Q.

A:

Q: Justify wether the following is an inner product or not =5x1y1-x1y2-x2y1+2x2y2 where x=(x1,x1) and…

A: By using axioms of inner product we have to show whether <x, y> is an inner product or not.

Q: Let T: R3 -> R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -> R2 Hence, T-1 : R2 ----> R3 So, necessarily we have to find a, b & c such that…

Q: A Show that the function f: R² → R defined by f(x₁, x₂) = x² + 2x² + 3x₁x₂ + 4x₁ + 5x₂ + 6 is a…

A: Definition of Vector Quadratic Function : Given Function f: R2→R is a vector quadratic function if…

Q: Prove that u × 0 = 0 × u = 0.

A:

Q: 3. Let A, B, T be bounded self-adjoint linear operators on a complex Hilbert space H. If T20 and…

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Q: R^n×n is invertible, show that AB is similar to BA for all B ∈ R^n×n.

A:

Q: Let C ~ Cauchy(0,1). Show that C² + C-2 =9 4C² + 2.

A: c^2+1/c^2 >2 thats why it is equal to 4c^2+2

Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: 17. Use Theorem 4.7 to prove that Vx + 3 2 V3 + x/4 for

A: Let,

Q: Let fi = 1+x + x² + x³ , f2 = 1 + 2x + 3x? + 6x³, f3 = 3+2x + ? + 5x4, and g = 1+x + 2a? + tæ³. For…

A:

Q: 4. Find x and y such that =x'y subject to *+*= 36, x20, y20 attains a maximum value. A) (24,12) B)…

A: Given: x+y=36, x≥0, y≥0   ……1   To find: x and y when z=x2y is maximum.

Q: 20. Prove that A is invertible if and only if A = QR, where Q is orthogonal and R is upper…

A: We have to  Prove that A is invertible if and only if A = QR, where Q is orthogonal and R is upper…

Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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Q: Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies.…

A:

Q: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis

Q: Suppose f is the function defined by f(x) = Ax, where a. The domain of f is b. The codomain of f is…

A: Dear student, as per our guidelines we can answer only first three subqustions. So kindly resubmit…