T: R R Let W be the plane in R given by 2x -z =...

  

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T: R³ → R³ Let W be the plane in R³ given by 2x -z = 0. Consider the linear transformation given by T(x) = projw(x). Note: You do not need to find the matrix A that induces this transformation to answer any part this question below. a. Describe Ker(T) geometrically, and give a basis for it. b. Describe Im(T) geometrically, and give a basis for it. c. State the eigenvalues and describe the corresponding eigenspaces of the matrix A that induces T. d. Explain why the matrix A that induces T must be symmetric. e. Orthogonally diagonalize A by stating a diagonal matrix D and an orthogonal matrix U such that A = UDUT. f. Is T an isomorphism? Justify your answer. T: R³ → R³ Let W be the plane in R³ given by 2x -z = 0. Consider the linear transformation given by T(x) = projw(x). Note: You do not need to find the matrix A that induces this transformation to answer any part this question below. a. Describe Ker(T) geometrically, and give a basis for it. b. Describe Im(T) geometrically, and give a basis for it. c. State the eigenvalues and describe the corresponding eigenspaces of the matrix A that induces T. d. Explain why the matrix A that induces T must be symmetric. e. Orthogonally diagonalize A by stating a diagonal matrix D and an orthogonal matrix U such that A = UDUT. f. Is T an isomorphism? Justify your answer.

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a Geometrically the kernel of T is the set of all vectors in R3 that are orthogonal to the plane W T... View the full answer