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Consider the plane P in R³ defined by the equation 2x + 3y+4z = 0. The transformation T : R³ → R³ maps a vector x to the closest vector T(x) lying in the plane P. In other words, 3 T: R³ R³ ХНУЕР where d(x, y) ≤ d(x, z) for all z Є P. This is a linear map (you don't need to prove this here, consider it a bonus exercise). Complete the following exercises: 1. Find a basis {b1, b2} for P. 2. Verify that T((2,3,4)) = (0,0,0) 3. Verify that B = {b1, b2, (2, 3, 4)} is a basis for R³ 4. Write the matrix (T)] B relative to the basis in part (c). 5. Write the matrix [T] 7 relative the the standard basis of R³. What is the rank and nullity of this matrix?

Consider the plane P in R³ defined by the equation 2x + 3y+4z = 0. The transformation T : R³ → R³ maps a vector x to the closest vector T(x) lying in the plane P. In other words, 3 T: R³ R³ ХНУЕР where d(x, y) ≤ d(x, z) for all z Є P. This is a linear map (you don't need to prove this here, consider it a bonus exercise). Complete the following exercises: 1. Find a basis {b1, b2} for P. 2. Verify that T((2,3,4)) = (0,0,0) 3. Verify that B = {b1, b2, (2, 3, 4)} is a basis for R³ 4. Write the matrix (T)] B relative to the basis in part (c). 5. Write the matrix [T] 7 relative the the standard basis of R³. What is the rank and nullity of this matrix?

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 13CR
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Consider the plane P in R³ defined by the equation 2x + 3y+4z = 0. The transformation T : R³ → R³ maps a vector x to the closest vector T(x) lying in the plane P. In other words, 3 T: R³ R³ ХНУЕР where d(x, y) ≤ d(x, z) for all z Є P. This is a linear map (you don't need to prove this here, consider it a bonus exercise). Complete the following exercises: 1. Find a basis {b1, b2} for P. 2. Verify that T((2,3,4)) = (0,0,0) 3. Verify that B = {b1, b2, (2, 3, 4)} is a basis for R³ 4. Write the matrix (T)] B relative to the basis in part (c). 5. Write the matrix [T] 7 relative the the standard basis of R³. What is the rank and nullity of this matrix?
Transcribed Image Text:Consider the plane P in R³ defined by the equation 2x + 3y+4z = 0. The transformation T : R³ → R³ maps a vector x to the closest vector T(x) lying in the plane P. In other words, 3 T: R³ R³ ХНУЕР where d(x, y) ≤ d(x, z) for all z Є P. This is a linear map (you don't need to prove this here, consider it a bonus exercise). Complete the following exercises: 1. Find a basis {b1, b2} for P. 2. Verify that T((2,3,4)) = (0,0,0) 3. Verify that B = {b1, b2, (2, 3, 4)} is a basis for R³ 4. Write the matrix (T)] B relative to the basis in part (c). 5. Write the matrix [T] 7 relative the the standard basis of R³. What is the rank and nullity of this matrix?
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