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Unformatted text preview: MA 35100 HOMEWORK ASSIGNMENT #5 SOLUTIONS Problem 1. pg. 110; prob. 24 Describe the images and kernels of the transformations in Exercises 23 through 25 geometrically: Orthogonal projection onto the plane x + 2 y + 3 z = 0. Solution: We may think of orthogonal projection as shining a flashlight from above onto the plane. Since we are shining onto an object, the image (i.e., surface illuminated) is the entire object: The image is the plane x + 2 y + 3 z = 0. To compute the kernel, we want to know those objects which cast no shadow when we shine this light. They are those objects which are orthogonal to the plane. The expression x +2 y +3 z is the dot product of the vectors ~v = 1 2 3 and ~x = x y z , so the plane of interest consists of all vectors ~x that are perpendicular to the vector ~v . By definition, ~v is a nonzero vector orthogonal to the plane, so any orthogonal projection is simply a projection onto the vector ~v . The kernel is the line spanned by the vector ~v = 1 2 3 . Problem 2. pg. 111; prob. 37 For the matrix A = 0 1 0 0 0 1 0 0 0 describe the images and kernels of the matrices A , A 2 , and A 3 geometrically. Solution: First we multiply out the matrices: A = 0 1 0 0 0 1 0 0 0 , A 2 = 0 0 1 0 0 0 0 0 0 , A 3 = 0 0 0 0 0 0 0 0 0 . 1 By Theorem 3.1.3 on page 105 of the text, the image of a matrix is the span of its column vectors. That means im( A ) = span 1 , 1 = span( ~e 1 , ~e 2 ) , im( A 2 ) = span 1 = span( ~e 1 ) , im( A 3 ) = = { ~ } . Geometrically, im( A ) is the x y plane in R 3 , im( A 2 ) is the xaxis, and im( A 3 ) is the origin. To compute the kernel of a matrix, we solve a system of equations: A~x = 0 1 0 0 0 1 0 0 0 x 1 x 2 x 3 = x 2 x 3 , A 2 ~x = 0 0 1 0 0 0 0 0 0 x 1 x 2 x 3 = x 3 . We see that A~x = ~ 0 when x 2 = x 3 = 0; A 2 ~x = ~ 0 when x 3 = 0, and A 3 ~x = ~ 0 for all ~x . This gives ker( A ) = span 1 = span( ~e 1 ) , ker( A 2 ) = span 1 , 1 = span( ~e 1 , ~e 2 ) , ker( A 3 ) = span 1 , 1 , 1 = R 3 . Geometrically, ker( A ) is the xaxis in R 3 , ker( A 2 ) is the x y plane, and ker( A 3 ) is all of 3dimensional space. Problem 3. pg. 111; prob. 38 Consider a square matrix A . a. What is the relationship between ker( A ) and ker( A 2 )? Are they necessarily equal? Is one of them necessarily contained in the other? More generally, what can you say about ker( A ), ker( A 2 ), ker( A 3 ), ker( A 4 ), ...?...
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 Spring '10
 EGoins
 Linear Algebra, Algebra, Transformations, Vector Space, 3 L, Ker, .

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