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33. Show that the transformation T defined by T(x1, x₂) (2x1 - 3x2, x₁ + 4, 5x₂) is not linear. =

33. Show that the transformation T defined by T(x1, x₂) (2x1 - 3x2, x₁ + 4, 5x₂) is not linear. =

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.6: Introduction To Linear Transformations
Problem 26EQ
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33. Show that the transformation T defined by T(x1, x2) (2x₁ - 3x2, x₁ + 4, 5x2) is not linear. 34. Let T: R" → R" be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation 7(x) = 0 has a nontrivial solution. [Hint: Suppose u and v in R" are linearly inde- pendent and yet T(u) and T(v) are linearly dependent. Then c₁T(u) + c₂T(v) = 0 for some weights c₁ and c₂, not both zero. Use this equation.] (S) 35. 3 Let T: R³ R³ be the transformation that reflects each vector x = (x1, X2, X3) through the plane x3 = 0 onto T(x) = (x₁, x2, -x3). Show that T is a linear transformation. [See Example 4 for ideas.] 36. Let T: R³ R3 be the transformation that projects each x2 25 vector x = (x1, x2, x3) onto the plane x₂ = 0, so T(x) = X(X₁, 0, x3). Show that T is a linear transformation. [M] In Exercises 37 and 38, the given matrix determines a linear transformation T. Find all x such that T(x) = 0. TiO 3 p x2 + Au X = 7 12 entl A 37. 39. 40 6) W [ h SOLUTIONS TO PRACTIC MIC 1377 1. A must have five colum codomain of T to be R². 2. Plot some random points
Transcribed Image Text:33. Show that the transformation T defined by T(x1, x2) (2x₁ - 3x2, x₁ + 4, 5x2) is not linear. 34. Let T: R" → R" be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation 7(x) = 0 has a nontrivial solution. [Hint: Suppose u and v in R" are linearly inde- pendent and yet T(u) and T(v) are linearly dependent. Then c₁T(u) + c₂T(v) = 0 for some weights c₁ and c₂, not both zero. Use this equation.] (S) 35. 3 Let T: R³ R³ be the transformation that reflects each vector x = (x1, X2, X3) through the plane x3 = 0 onto T(x) = (x₁, x2, -x3). Show that T is a linear transformation. [See Example 4 for ideas.] 36. Let T: R³ R3 be the transformation that projects each x2 25 vector x = (x1, x2, x3) onto the plane x₂ = 0, so T(x) = X(X₁, 0, x3). Show that T is a linear transformation. [M] In Exercises 37 and 38, the given matrix determines a linear transformation T. Find all x such that T(x) = 0. TiO 3 p x2 + Au X = 7 12 entl A 37. 39. 40 6) W [ h SOLUTIONS TO PRACTIC MIC 1377 1. A must have five colum codomain of T to be R². 2. Plot some random points
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