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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
33
![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-
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. 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) =
Z(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.
econil on
JETTE
SOLUTIONS TO PRAC
the be
1. A must have five col](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F787e472b-3dcf-45d7-8670-005f90d3000b%2F9e995a3f-a6b6-4e42-b1c0-d5089531b58e%2F7rvani_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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-
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. 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) =
Z(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.
econil on
JETTE
SOLUTIONS TO PRAC
the be
1. A must have five col
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