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. =
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)
(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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa091d57-466c-454e-b839-c6748796a9fc%2F9c83d7f5-5104-4ee5-af89-df2ef5d1ef5b%2Fo0ciaf6_processed.jpeg&w=3840&q=75)
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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