solutions-manual.51-60 - Exercises 51 p 1 4 u221a 2 u221a...

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Exercises 51 parenleftbigg 1 4 2 3 - 1 4 2 - 1 4 2 3 - 1 4 2 1 4 2 3 + 1 4 2 1 4 2 3 - 1 4 2 parenrightbigg Note that it doesn’t matter about the order in this case. 17. Find the matrix of the linear transformation which rotates every vector in R 3 counter clockwise about the z axis when viewed from the positive z axis through an angle of 30 and then reflects through the xy plane. a25 x a45 y a54 z 1 0 0 0 1 0 0 0 - 1 cos ( π 6 ) - sin ( π 6 ) 0 sin ( π 6 ) cos ( π 6 ) 0 0 0 1 = 1 2 3 - 1 2 0 1 2 1 2 3 0 0 0 - 1 18. Find the matrix for proj u ( v ) where u = (1 , - 2 , 3) T . Recall that proj u ( v ) = ( v , u ) | u | 2 u and so the desired matrix has i th column equal to proj u ( e i ) . Therefore, the matrix desired is 1 14 1 - 2 3 - 2 4 - 6 3 - 6 9 19. Find the matrix for proj u ( v ) where u = (1 , 5 , 3) T . As in the above, the matrix is 1 35 1 5 3 5 25 15 3 15 9 20. Find the matrix for proj u ( v ) where u = (1 , 0 , 3) T . 1 10 1 0 3 0 0 0 3 0 9 21. Show that the function T u defined by T u ( v ) v - proj u ( v ) is also a linear transformation. T u ( a v + b w ) = a v + b w - ( a v + b w · u ) | u | 2 u = a v - a ( v · u ) | u | 2 u + b w - b ( w · u ) | u | 2 u = aT u ( v ) + bT u ( w ) 22. Show that ( v - proj u ( v ) , u ) = 0 and conclude every vector in R n can be written as the sum of two vectors, one which is perpendicular and one which is parallel to the given vector. ( v - proj u ( v ) , u ) = ( v , u ) - parenleftBig ( v · u ) | u | 2 u , u parenrightBig = ( v , u ) - ( v , u ) = 0 . Therefore, v = v - proj u ( v ) + proj u ( v ) . The first is perpendicular to u and the second is a multiple of u so it is parallel to u . 23. Here are some descriptions of functions mapping R n to R n . (a) T multiplies the j th component of x by a nonzero number b. Saylor URL: The Saylor Foundation
52 Exercises (b) T replaces the i th component of x with b times the j th component added to the i th component. (c) T switches two components. Show these functions are linear and describe their matrices. Each of these is an elementary matrix. The first is the elementary matrix which multiplies the j th diagonal entry of the identity matrix by b . The second is the elementary matrix which takes b times the j th row and adds to the i th row and the third is just the elementary matrix which switches the i th and the j th rows where the two components are in the i th and j th positions. 24. In Problem 23, sketch the effects of the linear transformations on the unit square in R 2 . Give a geometric description of an arbitrary invertible matrix in terms of products of matrices of these special matrices in Problem 23. This picture was done earlier. Now if A is an arbitrary n × n matrix, then a product of these elementary matrices E 1 ··· E p has the property that E 1 ··· E p A = I. Hence A is the product of the inverse elementary matrices in the opposite order. Each of these is of the form in the above problem.

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