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.] transformation Tonto
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.] transformation Tonto
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
33
![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%2F7b4a148f-68e4-4f53-ad48-fc67cb19912c%2Foxcn9mo_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
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