Problem 6 (5+5=10 Marks). A plane in R is completely determined by any three distinct points...

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Problem 6 (5+5=10 Marks). A plane in R is completely determined by any three distinct points in it. Given any three distinct points u, v, w R the plane P passing through them is defined to be P= {au+by+cw | a+b+c=1, a, b, c R}. Let 9: R R be a map. Then show the following: 1. If g is linear, then P, := {(x, y, g(x, y)) | x, y R} is a plane. 2. If P = {(x, y, g(x, y)) | x, y = R} is a plane and g(0, 0) = 0, then g is a linear map. Problem 6 (5+5=10 Marks). A plane in R is completely determined by any three distinct points in it. Given any three distinct points u, v, w R the plane P passing through them is defined to be P= {au+by+cw | a+b+c=1, a, b, c R}. Let 9: R R be a map. Then show the following: 1. If g is linear, then P, := {(x, y, g(x, y)) | x, y R} is a plane. 2. If P = {(x, y, g(x, y)) | x, y = R} is a plane and g(0, 0) = 0, then g is a linear map.

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1 To show that if g is linear then Pg x y gxy xy R is a plane we need to show that Pg can be describ... View the full answer