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.
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.
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
29
![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
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.
a
[2]
0
d
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/;
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. TR4 → 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 Rm 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" → Rm
be a linear transformation such that T(x) = Bx for some
34. Why is the question "Is the linear transformation T onto?"
an existence question?
35. If a linear transformation T: R"R" maps R" onto Rm
can you give a relation between m and n? If T is one-to-one,
what can you say about m and n?
mxn matrix B. Show that if A is the standard matrix for
T, then A= B. [Hint: Show that A and B have the same
columns.]
36. Let S : RP→ R" and T: R"→ R" be linear transforma-
tions. Show that the mapping x→ T(S(x)) is a linear trans-
formation (from RP to Rm). [Hint: Compute T(S(cu + dv))
for u, v in RP and scalars c and d. Justify each step of the
computation, and explain why this computation gives the
desired conclusion.]
[M] In Exercises 37-40, let T be the linear transformation whose
standard matrix is given. In Exercises 37 and 38, decide if T is a
one-to-one mapping. In Exercises 39 and 40, decide if T maps R5
onto R5. Justify your answers.
1024
tait
37.
39.
40.
-5
10 -5 4
8
7
3-4
4-9 5 -3
5 4
-3-2
4 -7 3 7 5
6-8 5 12 -8
10 -8 -9 14
3 -5 4 2 -6
6 -6
-7
3
котор адфра
-7
9 13
6
5 6
15 -7 -6
-8 -9
-5 -6 -89
14
-1
4
12 -5 -9
8
14 15 2 11
13
38.
Г 7
5
10
6
12 8 12
-8
-6-2
9 ini ( 2mol
adb davoni ziniog abal
SA
191
4 -9
16 -4
7
5
Jonsd) ba
tedi
1600 10](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7aff33-da2d-447a-b7cd-4f086825b0f0%2F7a51dc03-97e8-4c5f-b66d-165df5ee34ab%2Fwvvg77a_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
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.
a
[2]
0
d
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/;
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. TR4 → 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 Rm 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" → Rm
be a linear transformation such that T(x) = Bx for some
34. Why is the question "Is the linear transformation T onto?"
an existence question?
35. If a linear transformation T: R"R" maps R" onto Rm
can you give a relation between m and n? If T is one-to-one,
what can you say about m and n?
mxn matrix B. Show that if A is the standard matrix for
T, then A= B. [Hint: Show that A and B have the same
columns.]
36. Let S : RP→ R" and T: R"→ R" be linear transforma-
tions. Show that the mapping x→ T(S(x)) is a linear trans-
formation (from RP to Rm). [Hint: Compute T(S(cu + dv))
for u, v in RP and scalars c and d. Justify each step of the
computation, and explain why this computation gives the
desired conclusion.]
[M] In Exercises 37-40, let T be the linear transformation whose
standard matrix is given. In Exercises 37 and 38, decide if T is a
one-to-one mapping. In Exercises 39 and 40, decide if T maps R5
onto R5. Justify your answers.
1024
tait
37.
39.
40.
-5
10 -5 4
8
7
3-4
4-9 5 -3
5 4
-3-2
4 -7 3 7 5
6-8 5 12 -8
10 -8 -9 14
3 -5 4 2 -6
6 -6
-7
3
котор адфра
-7
9 13
6
5 6
15 -7 -6
-8 -9
-5 -6 -89
14
-1
4
12 -5 -9
8
14 15 2 11
13
38.
Г 7
5
10
6
12 8 12
-8
-6-2
9 ini ( 2mol
adb davoni ziniog abal
SA
191
4 -9
16 -4
7
5
Jonsd) ba
tedi
1600 10
Transcribed Image Text:1. a.
1
0
000
1 0 0
1.2
EXERCISES
In Exercises 1 and 2, determine which matrices are in reduced
echelon form and which others are only in echelon form.
001
4. Write the system of equations corre
5. Rewrite each nonzero equation fro
expressed in terms of any free vari
PRACTICE PROBLEMS
1. Find the general solution of the line
[1
2. Find the general solution of the sy:
BERT
BROEDSL WOR
b.
3. Suppose a 4 x 7 coefficient mat
system consistent? If the system
1
0
0
07
0
1
0 0
1
1 0
X1 -
-2x1 +
3x1 -
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