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
![In Exercises 25-28, determine if the specified linear transfor
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
01
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. ats
TANGA
olisy
29. T:R³ → R4 is one-to-one.
30. T:R4 → R³ is onto.
31. Let T: R" → Rm 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
columns." Explain why the statement is true. [Hint: Look in
the exercises for Section 1.7.
pivot
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" → Rm
be a linear transformation such that T(x) = Bx for some
WEB
o
x2
[M]
stand
one-
onto
37.
39.
40.
SOLUTION TO PRACTICE F
1. Follow what happens to e, a
and then is reflected into -e
shear transformation. Since](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5495aa8f-e580-46c5-af54-23236303b730%2F485c5415-0f22-4c7d-8ee3-8a78a700bc26%2Ferhvbl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Exercises 25-28, determine if the specified linear transfor
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
01
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. ats
TANGA
olisy
29. T:R³ → R4 is one-to-one.
30. T:R4 → R³ is onto.
31. Let T: R" → Rm 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
columns." Explain why the statement is true. [Hint: Look in
the exercises for Section 1.7.
pivot
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" → Rm
be a linear transformation such that T(x) = Bx for some
WEB
o
x2
[M]
stand
one-
onto
37.
39.
40.
SOLUTION TO PRACTICE F
1. Follow what happens to e, a
and then is reflected into -e
shear transformation. Since
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