33. Show that the transformation T defined by T(x1, x₂) (2x₁3x2, 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-mol 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.] 35. 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 2 vector x = (x₁, x2, x3) onto the plane x2 = 0, so T(x) =m Z(X1, 0, x3). Show that T is a linear transformation. 21012 [M] In Exercises 37 and 38, the given matrix determines a linear

33. Show that the transformation T defined by T(x1, x₂) (2x₁3x2, 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-mol 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.] 35. 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 2 vector x = (x₁, x2, x3) onto the plane x2 = 0, so T(x) =m Z(X1, 0, x3). Show that T is a linear transformation. 21012 [M] In Exercises 37 and 38, the given matrix determines a linear

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.CR: Review Exercises

Problem 65CR

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