Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. i) For any non zero x in X, there is a bounded linear functional in X such that ||||= 1 and (x) = ||x||. ii) If x, y in X verify 4(x) = (y) in X', then x = y.

Q: Determine whether T is a linear transformation. T: F --->R defined by               T(f) = f( c),…

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Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function f(r) = VI+ tª dt.

A: We will use the first fundamental theorem of calculus to evaluate the derivative.   The detailed…

Q: Let R = {(x,y): |x| + |y| ≤ 2}. Use the transformation x=u+v, y=u-v to evaluate ∫∫ ((x^2)*(e^y))dA

A: Consider the given: R=x,y:x+y≤2x=u+vy=u−vA=111−1=−1−1=−2=2

Q: Let f:  V-->W be any linear transformation. What are possible solutions to f(0)?

A: Note: A Linear Transformation Maps the Zero vector to the Zero Vector.

Q: Which of the following mappings A: R^2->R is a linear functional? (a) (a,b) -> lal + lbl. (b)…

A: Step 1:A linear functional is a linear map from a vector space to its underlying field. Formally,…

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Q: Show that the linear transformation is continuous at every Xo = R³. F(X) = x₁ + x2 + x3 2x13x2 + x3…

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Q: Find the transitive closure of R={(0,1),(0,2),(1,1), (1,3),(2,2),(3,0)}

A: Given relation is                              R=(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0) we…

Q: Show f(x) = e and g(x) = xe linearly independent by finding its Wronskian.

A: Given: fx=exgx=xex The formula to obtain the Wronskian is Wx=f(x)g(x)f'(x)g'(x), where f(x) and g(x)…

Q: Let f(x, y, z) = x²y³ + zª and x = s², y = st², and z = (a) Calculate the primary derivatives - af…

A: fx, y, z=x2y3+z4  &  x=s2, y=st2,  &  z=s2t a). ∂f∂x=∂∂xx2y3+z4=y3·2x+0=2xy3…

Q: THEOREM 4.4 (LINEAR TRANSLATION) Let Y= aX + b; then: My(0) = eb" My(a0).

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Q: The mapping L(x) = ax + b is linear. ihi O

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Q: əz əz 1. Let z = x² + y³, x = 2st and y = s − t². Compute for and əx Ət

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Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: Q2: Let T₁, T₂, be the following bounded linear operators ¹ → T₁(x₁, x₂. x3,) = (x₁, X₁, X₁, X₁,…

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Q: Let L:R→R be a linear operator, if L(X)= L(Y), then X=Y. Select one: O True O False

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Q: . Let B(X, Y) denote the space of all bounded linear operators between the normed spaces X and Y.…

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Q: Find the null space of the linear transformation T: R →R TE, y, z) = (x + y, z). defined by

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A: Given that  be a linear map and .To find the value of .

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Q: If F(t) = 2xyi + xyz2j + ex+yk, then what is the divergence of F? Select one: A. 2yi + xz2j + ex+yk…

A: The divergence of a vector F=P,Q,R is given by div F=∂P∂x+∂Q∂y+∂R∂z. The given vector is Ft=2xy…

Q: use implici't differen tiatiom to brind dyldu at the pont (2,2) = 1+y" エ+Scu

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Q: 17. Let y₁ = x₁ + x₂ and y₂ = x² + x² + 2(x₁ + x₂ + x₁x₂). (a) Use the Jacobian to test whether…

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Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

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Q: EHET X1 X2 - X1 Let L:R3 → R³ be a linear operator such that L( x2 ) =| x3- X3 - X2 , then Ker(L)…

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