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Computing Jacobians Compute the Jacobian J(u, ν) for the following transformation.
T: x = (u + ν)/√2, y = (u - ν)/√2
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- Calculus Define T:P2R by T(p)=01p(x)dx What is the kernel of T?CAPSTONE Let T:R4R3 be the linear transformation represented by T(x)=Ax, where A=[121001230001]. (a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T is onto? Explain. (f) Is T an isomorphism? Explain.Let T:P2P3 be the linear transformation T(p)=xp. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3}.
- Let T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.Guided Proof Let B be an invertible nn matrix. Prove that the linear transformation T:Mn,nMn,n represented by T(A)=AB is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} Theorem 6.6. Use the fact that B is an invertible nn matrix and that T(A)=AB. (iii) Conclude that T is an isomorphism.Computing Jacobians Compute the Jacobian J(u, ν) for the following transformation. T: x = 2uν, y = u2 - ν2
- Computing Jacobians Compute the Jacobian J(u, ν) for the following transformation. T: x = u cos πν, y = u sin πνJacobians in three variables Evaluate the Jacobians J(u, ν, w) for the following transformation. x = ν + w, y = u + w, z = u + νComputing Jacobians Compute the Jacobian J(u, ν) of the followingtransformation. T: x = u + ν, y = u - ν
- Computing Jacobians Compute the Jacobian J(u, ν) of the followingtransformation. T: x = 3u, y = 2ν + 2A linear transformation Tis determined when we know T(i), T(j), T(k). For an affine transformation we also need T( __ ). The input point ( x, y, z, 1) is transformed to xT(i) + yT(j) + zT(k) + __ .Given the linear transformation below: 1. Determine the transformation matrix of the linear transformation above2. Determine Ker(T) and R(T)