Define T: C[0, 1] → C[0, 1] by 1 Tf(x) = sin (x - y) f(y) dy: Show that the range of T is finite dimensional. Is T a compact operator? Justify your answer.

Q: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: 1. Determine the domain and range of the vector-valued function r(t) = (√t+3, In(2 – t), t²)

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Q: 3. Find the fixed point(s) of f : [1, 0) → [2, 0) defined by f(x) = vx² + x.

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Q: Show that the operator U is positive U: 1² ([0,1]) → L²([0,1]), Uƒ (x) = X[{0,1] [~*\f(t)\dt

A: To show that the operator U is positive, need to demonstrate that for any f∈L2([0,1]), the inner…

Q: Draw a picture on the z-plane

A: In this question, we draw a picture of the given curve in C(t). here, first, we write the real and…

Q: Let f : X → Y and g : X → Z be two bijective functions. Is the function h: X → Y × Z defined by h(x)…

A: Given: f:X→Y and  g:Y→Z be two bijective functions. To Check: Whether the function h:X→Y×Z defined…

Q: Find the adjoint of F:C3 tends to C3 defined by F(z1, z2, z3)=(2z1+(1-i)z2 , (3+2i)z1 -4iz3 ,…

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Q: Find and sketch the domain of z = f(x, y) = In(9 - x - y²).

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Q: anction of z = x + iy

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Q: Find the analytic function f (z)u+iv if u = a (1+cose).

A: We have to find the analytic function

Q: Let F(x, y, z) = (14xyz + 7sin(x))i + 7x²zj + 7x²yk. Find a function f such that F = Vf. f(x, y, z)…

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Q: Find and sketch the domain of the function f(x,y) = cosx + V3x-y+2

A: The given function is .The objective is to find the domain of the function and sketch the domain.

Q: A = [a,b), be an (right half-open) interval of R. 6 (c) (d) a,be R+,b+00, Explain whether f(A) is…

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Q: 2. Let f= L(X,Y). Show that ƒ is one to one iff (f(x).(x)....(x)) is linear independent when (x,x)…

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Q: Let (t) , a) Find the domain of F(t) b) Find " (t)

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Q: Find all functions f:R → R that satisfy the equality xf(y) + yf (x) = (x+ y)f(x)f(y) %3D for all r,…

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Q: Sketch the graph of the vector-valued function F(t) = (2t – 1)² î +(2t +2) ĵ. Draw arrows on your…

A: The vector valued functions in two dimensions can be represented as r→(t)=x(t)i^+y(t)j^.

Q: C: r(t) = (t2 + 4)i + (t2 – 1)j + (t2 – 2t)k, 0sts 2 (a) Find a function f such that F = Vf. f(x, y,…

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Q: Let F = (2xz+y², 2xy, x²+1), if Vf= F, find f(1, 1, 1) – f(0, 0, 0)

A: Given vector funtion and

Q: 2. Consider the function f : R²R such that f(x, y) = x²y-2xy² +6xy-3y² +18y. (a) Show that f has…

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Q: Consider the function f: N× N → Q given by f(x, y) = your claim. X y + 1 Find the image of f and…

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