25. The transformation in Exercise 17
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
25
![80 CHAPTER 1 Linear Equations in Linear Algebra
c. The standard matrix of a linear transformation from R²
to R2 that reflects points through the horizontal axis,
the vertical axis, or the origin has the form [8 2].
where a and d are ±1.
d. A mapping T: R"R" is one-to-one if each vector in
R" maps onto a unique vector in Rm.
e. If A is a 3 x 2 matrix, then the transformation x → Ax
cannot map R2 onto R³.
In Exercises 25-28, determine if the specified linear transforma-
tion is (a) one-to-one and (b) onto. Justify each answer.
25. The transformation in Exercise 17
26 The transformation in Exercise 2
27. The transformation in Exercise 19
28. The transformation in Exercise 14
In Exercises 29 and 30, describe the possible echelon forms of the
standard matrix for a linear transformation T. Use the notation of
Example 1 in Section 1.2.
29. T: R³
R4 is one-to-one.
30. T: R4
R³ is onto.
31. Let T: R" → R" be a linear transformation, with A its
standard matrix. Complete the following statement to make
it true: "T is one-to-one if and only if A has - pivot
columns." Explain why the statement is true. [Hint: Look in
the exercises for Section 1.7.]
32. Let T: R" → R" be a linear transformation, with A its
standard matrix. Complete the following statement to make
it true: "T maps R" onto R" if and only if A has
pivot columns." Find some theorems that explain why the
statement is true.
33. Verify the uniqueness of A in Theorem 10. Let T: R → R
be a linear transformation such that T(x) = Bx for some
[
S
с
C
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F973d6112-791f-4249-91d8-33d0c69fc61e%2F040c706f-a74b-44e8-950c-eb2679650726%2F2iztqoo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:80 CHAPTER 1 Linear Equations in Linear Algebra
c. The standard matrix of a linear transformation from R²
to R2 that reflects points through the horizontal axis,
the vertical axis, or the origin has the form [8 2].
where a and d are ±1.
d. A mapping T: R"R" is one-to-one if each vector in
R" maps onto a unique vector in Rm.
e. If A is a 3 x 2 matrix, then the transformation x → Ax
cannot map R2 onto R³.
In Exercises 25-28, determine if the specified linear transforma-
tion is (a) one-to-one and (b) onto. Justify each answer.
25. The transformation in Exercise 17
26 The transformation in Exercise 2
27. The transformation in Exercise 19
28. The transformation in Exercise 14
In Exercises 29 and 30, describe the possible echelon forms of the
standard matrix for a linear transformation T. Use the notation of
Example 1 in Section 1.2.
29. T: R³
R4 is one-to-one.
30. T: R4
R³ is onto.
31. Let T: R" → R" be a linear transformation, with A its
standard matrix. Complete the following statement to make
it true: "T is one-to-one if and only if A has - pivot
columns." Explain why the statement is true. [Hint: Look in
the exercises for Section 1.7.]
32. Let T: R" → R" be a linear transformation, with A its
standard matrix. Complete the following statement to make
it true: "T maps R" onto R" if and only if A has
pivot columns." Find some theorems that explain why the
statement is true.
33. Verify the uniqueness of A in Theorem 10. Let T: R → R
be a linear transformation such that T(x) = Bx for some
[
S
с
C
3
![(4,-7), and
columns of the
through 37/2
through -л/4
-1/√2).]
on that maps ei
ged.
ation that leaves
-3л/4 radian
n the horizontal
e horizontal x₁-
x2 = X1.
hear that trans-
ed) and then re-
vertical x2-axis
= reflects points
through the X2-
a linear transfor-
What is the angle
is merely a rota-
e rotation?
in such that T(e₁)
gure. Using the
In Exercises 15 and 16, fill in the missing entries of the matrix,
assuming that the equation holds for all values of the variables.
?
15. ?
?
16.
?
?
?
?
?
?
?
?
?
X1
X2
X3
?
X1
JCH-[
?
X2
?
3x₁2x3
4x1
x1 - x₂ + x3
T(x₁, x2) =
(3,8).
XI - X2
-2x1 + x₂
X1
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x₂,...
are not vectors but are entries in vectors.
17. T(X1, X2, X3, X4) = (0, X₁ + X2, X₂ + X3, X3 + x4)
18. T(x₁, x₂) = (2x2 – 3x₁, x₁ - 4x2, 0, x₂)
19.
T(X1, X2, X3) = (x₁ - 5x2 + 4x3, x2 - 6x3)
20. T(x1, x2, x3, x4) = 2x1 + 3x3 - 4x4
(T: R4 → R)
21. Let T: R2 R2 be a linear transformation such that
(x₁ + x2, 4x1 + 5x₂). Find x such that T(x) =
22. Let T: R2 R³ be a linear transformation such that
T(x₁, x₂) = (x₁2x2, -x₁ + 3x2, 3x1 - 2x₂). Find x such
that T(x)= (-1,4,9).
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
23. a. A linear transformation T: R" → R" is completely de-
termined by its effect on the columns of the n x n identity
matrix.
b. If T: R² R2 rotates vectors about the origin through
an angle , then T is a linear transformation.
c. When two linear transformations are performed one after
another, the combined effect may not always be a linear
transformation.
d A manning T R"R" is onto R if every yectory in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F973d6112-791f-4249-91d8-33d0c69fc61e%2F040c706f-a74b-44e8-950c-eb2679650726%2F9v4fhd8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(4,-7), and
columns of the
through 37/2
through -л/4
-1/√2).]
on that maps ei
ged.
ation that leaves
-3л/4 radian
n the horizontal
e horizontal x₁-
x2 = X1.
hear that trans-
ed) and then re-
vertical x2-axis
= reflects points
through the X2-
a linear transfor-
What is the angle
is merely a rota-
e rotation?
in such that T(e₁)
gure. Using the
In Exercises 15 and 16, fill in the missing entries of the matrix,
assuming that the equation holds for all values of the variables.
?
15. ?
?
16.
?
?
?
?
?
?
?
?
?
X1
X2
X3
?
X1
JCH-[
?
X2
?
3x₁2x3
4x1
x1 - x₂ + x3
T(x₁, x2) =
(3,8).
XI - X2
-2x1 + x₂
X1
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x₂,...
are not vectors but are entries in vectors.
17. T(X1, X2, X3, X4) = (0, X₁ + X2, X₂ + X3, X3 + x4)
18. T(x₁, x₂) = (2x2 – 3x₁, x₁ - 4x2, 0, x₂)
19.
T(X1, X2, X3) = (x₁ - 5x2 + 4x3, x2 - 6x3)
20. T(x1, x2, x3, x4) = 2x1 + 3x3 - 4x4
(T: R4 → R)
21. Let T: R2 R2 be a linear transformation such that
(x₁ + x2, 4x1 + 5x₂). Find x such that T(x) =
22. Let T: R2 R³ be a linear transformation such that
T(x₁, x₂) = (x₁2x2, -x₁ + 3x2, 3x1 - 2x₂). Find x such
that T(x)= (-1,4,9).
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
23. a. A linear transformation T: R" → R" is completely de-
termined by its effect on the columns of the n x n identity
matrix.
b. If T: R² R2 rotates vectors about the origin through
an angle , then T is a linear transformation.
c. When two linear transformations are performed one after
another, the combined effect may not always be a linear
transformation.
d A manning T R"R" is onto R if every yectory in
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