29 25. The transformationin Exercise 1726 The transformationin Exercise 227. The transformation in Exercise 1928. The transformation in Exercise 14In Exercises 29 and 30, describe the possible echelon forms of thestandard matrix for a linear transformation T. Use the notation ofExample 1 in Section 1.2.29. T: R³30. TR4 R³ is onto.31. Let T: R" → Rm be a linear transformation, with A itsstandard matrix. Complete the following statement to makeit true: "T is one-to-one if and only if A haspivotcolumns." Explain why the statement is true. [Hint: Look inthe exercises for Section 1.7.]CHR4 is one-to-one.32. Let T: R" → Rm be a linear transformation, with A itsstandard matrix. Complete the following statement to makeit true: "T maps R" onto R" if and only if A haspivot columns." Find some theorems that explain why thestatement is true.33. Verify the uniqueness of A in Theorem 10. Let TR" → Rmbe a linear transformation such that T(x) = Bx for someWEBstaoron'10!373940.SOLUTION TO PRACTICE F1. Follow what happens to e, aand then is reflected into -eshear transformation. Since noted DEFINITION1) 101 1A sau smo mot noroninoSear1THEOREM 1A rectangular matrix is in echelon form (or row echelon form) if it has thefollowing three properties:1. All nonzero rows are above any rows of all zeros.2. Each leading entry of a row is in a column to the right of the leading entry ofthe row above it.3. All entries in a column below a leading entry are zeros.If a matrix in echelon form satisfies the following additional conditions, then it isin reduced echelon form (or reduced row echelon form):4. The leading entry in each nonzero row is 1.5. Each leading 1 is the only nonzero entry in its column.odsisool bassol holadas additional examples.An echelon matrix (respectively, reduced echelon matrix) is one that is in echelonform (respectively, reduced echelon form). Property 2 says that the leading entries forman echelon ("steplike") pattern that moves down and to the right through the matrix.Property 3 is a simple consequence of property 2, but we include it for emphasiz.The "triangular" matrices of Section 1.1, such asandogl 1.2 Row Reduction and Echelon Forms000*2-3 2 10 1-4 8HOT0 005/2are in echelon form. In fact, the second matrix is in reduced echelon form. Here are**000 0**00qulogEXAMPLE 1 The following matrices are in echelon form. The leading entries (1)may have any nonzero value; the starred entries (*) may have any value (including zero).1 0 ***0 1 *000 0000000000 0and00000*00000 001000*100■00*0000201001 00 1■0Uniqueness of the Reduced Echelon Form***2916■000000100010001000000*3*****The following matrices are in reduced echelon form because the leading entries are 1's,and there are 0's below and above each leading 1.***0 ■*****0**0000113*********Any nonzero matrix may be row reduced (that is, transformed by elementary rowoperations) into more than one matrix in echelon form, using different sequences of rowoperations. However, the reduced echelon form one obtains from a matrix is unique. Thefollowing theorem is proved in Appendix A at the end of the text.o aquivalent to one and only one reduced echelon matrix.

For a linear transformation T: R^3 -> R^4 to be one-to-one, the standard matrix of T must have…

Answered: In Exercises 29 and 30, describe the…

In Exercises 29 and 30, describe the possible echelon forms of t 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.

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Chapter1: Introduction

Section: Chapter Questions

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Question

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
30. TR4 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
pivot
columns." Explain why the statement is true. [Hint: Look in
the exercises for Section 1.7.]
CH
R4 is one-to-one.
32. Let T: R" → Rm 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 TR" → Rm
be a linear transformation such that T(x) = Bx for some
WEB
sta
or
on
'10!
37
39
40.
SOLUTION TO PRACTICE F
1. Follow what happens to e, a
and then is reflected into -e
shear transformation. Since

noted DEFINITION
1) 101 1A sau smo mot no
ronino
Sear
1
THEOREM 1
A rectangular matrix is in echelon form (or row echelon form) if it has the
following three properties:
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of
the row above it.
3. All entries in a column below a leading entry are zeros.
If a matrix in echelon form satisfies the following additional conditions, then it is
in reduced echelon form (or reduced row echelon form):
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
odsisool bassol holadas additional examples.
An echelon matrix (respectively, reduced echelon matrix) is one that is in echelon
form (respectively, reduced echelon form). Property 2 says that the leading entries form
an echelon ("steplike") pattern that moves down and to the right through the matrix.
Property 3 is a simple consequence of property 2, but we include it for emphasiz.
The "triangular" matrices of Section 1.1, such as
andogl 1.2 Row Reduction and Echelon Forms
0
0
0
*
2-3 2 1
0 1-4 8
HOT
0 00
5/2
are in echelon form. In fact, the second matrix is in reduced echelon form. Here are
*
*
00
0 0
*
*
0
0
qulog
EXAMPLE 1 The following matrices are in echelon form. The leading entries (1)
may have any nonzero value; the starred entries (*) may have any value (including zero).
1 0 *
*
*
0 1 *
000 0
000
0
0
0
0
0
0 0
and
0
0
0
0
0
*
0
0
0
0
0 0
0
1
0
0
0
*
1
0
0
■
0
0
*
0
0
0
0
201
0
0
1 0
0 1
■
0
Uniqueness of the Reduced Echelon Form
*
*
*
29
16
■
000
0
00
1
0
0
0
1
0
0
0
1
000000
*
3
*
*
*
*
*
The following matrices are in reduced echelon form because the leading entries are 1's,
and there are 0's below and above each leading 1.
*
*
*
0 ■
*
*
*
*
*
0
*
*
0
0
0
0
1
13
*
*
*
*
*
*
*
*
*
Any nonzero matrix may be row reduced (that is, transformed by elementary row
operations) into more than one matrix in echelon form, using different sequences of row
operations. However, the reduced echelon form one obtains from a matrix is unique. The
following theorem is proved in Appendix A at the end of the text.
o aquivalent to one and only one reduced echelon matrix.

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