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# hw4 - Math 260 Spring 2012 Jerry L Kazdan Problem Set 4 Due...

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Math 260, Spring 2012 Jerry L. Kazdan Problem Set 4 Due: Never [Exam 1 is on Thursday, Feb.9, 12:00-1:20] Unless otherwise stated use the standard Euclidean norm . 1. In R 2 , define the new norm of a vector V = ( x, y ) by V := 2 | x | + | y | . Show this satisfies the properties of a norm. [ Remark: This is a “taxicab” norm where, because of traffic it is twice as expensive to go across town than up town.] 2. a) In R 3 , find the distance from the point (1 , 1 , 1) to the plane x + 2 y - z = 3. b) In R 4 , compute the distance from the point (1 , - 2 , 0 , 3) to the hyperplane x 1 + 3 x 2 - x 3 + x 4 = 3. 3. a) In R 3 , find an orthogonal basis for the plane x + 2 y - z = 0. b) Use it to find the orthogonal projection of V = (1 , 0 , - 1) into this plane. c) What is the orthogonal projection of V perpendicular to this plane? 4. In R 5 ,write X = ( x 1 , x 2 , x 3 , x 4 , x 5 ) and let S be the subspace spanned by the vectors V 1 := (1 , 0 , 0 , 0 , 0) , V 2 := (0 , 3 , 4 , 0 , 0) , V 3 := (0 , 4 , - 3 , 0 , 0) . Find the orthogonal projection P S ( X ) of X into S and the projection of X orthogonal to S .

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hw4 - Math 260 Spring 2012 Jerry L Kazdan Problem Set 4 Due...

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