homework_5_solutions

# homework_5_solutions - MA 35100 HOMEWORK ASSIGNMENT#5...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 x-axis, 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 x-axis in R 3 , ker( A 2 ) is the x- y plane, and ker( A 3 ) is all of 3-dimensional 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 ), ...?...
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

homework_5_solutions - MA 35100 HOMEWORK ASSIGNMENT#5...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online