Let f(x) = + 3x? + 4x. a) What is the kernel of f? b) Compute the minimum and the maximum of f.

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A: y=2(x-1)(x+5) first multiply (x-1)(x+5) =>(x-1)(x+5) =>(x^2 +5x -x -5) =>(x^2 +4x-5)

Q: Let f(x) = 3x² and. P = {–2, –1, 0, 1, 2, 3}. Compute Lf(P) and U,(P). %3D %3D

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A: To prove that linear maps are bounded.

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A: Solution is given below:

Q: Verify the Cauchy-Schwarz Inequality for u = (1, −1, 3) and v = (2, 0, −1).

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Q: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: Show that the norm of the maximum satisfies the definition of norm

A: The objective is to show that the norm of the maximum satisfies the definition of norm. Consider the…

Q: 5. Let f(z) = 3z* + iz3 + z + 7. Prove that all roots of f (z) are in the annulus 1 < |z| < 2.

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Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

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A: Here the Hamacher product is the t-norm given by t(a,b)=aba+b-ab We have to show that this function…

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Q: Do the three planes x1 + 2x2 + x3 = 4, x2 – x3 = 1, and x1 + 3x2 = 0 have at least one common point…

A: The three planes x1+2x2+x3=4, x2-x3=1, and x1+3x2=0.

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Q: let f (z) = z2 + z find the maximum value of If(z)l A = {z € C:0 < x, y < 2} on the square

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Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: 7. Determine whether (u,v) =3u, v, – 2u, v, is a valid inner-product function or not for R²

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Q: Determine a scalar r such that Ax = rx, where

A: Matrix: A=12-11014-45x=-12141AX=12-11014-45.-12141

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A: Solution is here

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A: Given,

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Q: b.) answer. Let f = x² and g = (1-x)². Are x and y orthogonal in this inner product space? Justify…

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Q: 9. verify that lx1l= max ilt)| defines. telasb] a on the Space Ccla, b]. morm Solution :-

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Q: 2. Let a,, az,.., a, be numbers. Show that there is a single number r such that foxo: (-a,) + (z –…

A: Solving

Q: Let f(x, y, z) = x2 (e)^y/z. Compute f(-1, 0, 1), f(1, 3, 3), andf(5, -2, 2).

A: Let fx,y,z=x2eyz 1. we have to compute f(-1,0,1) as, fx,y,z=x2eyzHere x=-1, y=0,z=1f-1,0,1=-12e01=1…

Q: Let f(z) = 3-1교2-8고 + 쯤. a) Compute the minimum and the maximum of f.

A: f(x)=19x3-13x2-83x+269

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Q: Find the linear least squares polynomial approximation to f(x) on the indicated interval if a. f(x)…

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Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

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Q: Assume that f:R² → R² is a continuous bijection such that the image of each line is a line. Show…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let f(x,y) = x - 5xy -y². Compute f(6,0), f(6, – 2), and f(b,c).

A: Solved below

Q: Show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). ⟨u, v⟩ =…

A: Given function u,v=u1v1+9u2v2 where u=u1,u2 and v=v1,v2 The function is defined on R2. To show that…

Q: let f(z) = z2 + z find the maximum value of If(z)| on the square A = {z € C:0 < x,y < 2}

A: We have given function fz=z2+z where, z=x+iy complex number We have to find maximum value of fz on…

Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2