Theorem 1 Every Pythagorean triple has the form (d(p2-q²), 2dpq, d(p²+q²)) for some integers p and q with p>q> 0 and some positive rational number d.

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Q: 7. Please show all equations

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A: Required python code:# Define the continuous mapping T: [0,1]^2 -&gt; [0,1]^2 def T(x, y):…

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A: Step 1:Step 2:Step 3:

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A: Required code:# Define the functions u(x) = 2x and v(x) = 1 - x over the interval [0,1] u_vals = 2 *…

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A: Use this python code:# Re-plot the original vectors and their images under the duality mapping…

Q: Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs…

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A: Approach to solving the question: Detailed explanation: Examples: Key references:Philosophy

Q: Prove that if AC BCR, then A*(A) ≤ X*(B).

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Q: u(x,t) = [* H(x − y, t)f(y) dy = e-(x-y)²/4t -f(y) dy, xER, t> 0. (23.6) With this explicit formula…

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A: For b.  Step 1: Form the Augmented MatrixStep 2: Multiply first row by -1 to make the pivot element…

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Q: Answer questions three only and it's a different equation problem show all work

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

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A: Step 1:

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Q: Use the inverse of matrix A to solve the following matrix-vector equation, Arb: 1 71 [4] - Do all…

A: Step 1: Given that, solve the given matrix-vector equation using the inverse of matrix A.…

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A: Step 1    Step 2

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A: 3.4.

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Q: 1. A space-ship is heading towards a planet, following the trajectory, -t r(t) = Ae¯² (5 cos(4t), 4…

A: Step 1:Position of the spaceship as a function of time:r(t)=Ae−t(5cos(4t),4sin(4t),−3sin(4t)) where…