1. Let A = , and define T : R² → R² by T(x) = Ax. -6 10 %3D 2 -4 10. A = Find the images under T of u = 3 1 and v = 4. 10 8 3 2. Let A = a. u = and v = 11. Let b = and let A be the Define T : R in the range of the linear transforr why not? R' by T(x) = Ax. Find T(u) and T(v). %3D In Exercises 3-6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. %3D 12. Let b = and let A be the 0 -3 6 ,b = 1 -1 3. A = -3 1 3 b in the range of the linear transfor why not? 2 -2 -1 1 -2 1 -3 ,b = 3 4. A = In Exercises 13-16, use a rectangular -5 6. u = V = and their images 1 -5 mation T. (Make a separate and reason exercise.) Describe geometrically wha in R?. 5. A = -3 %3D 7 1 -3 27 8 ,b 13. T(x) = 3 -8 6. A = %3D 1 3 1 8. 10 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: R -→ R° by T(x) = Ax? 15. T(x) = 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R’ by the rule T(x) = Ax? 16. T) =: :: 16. T(x): For Exercises 9 and 10, find all x in R that are mapped into the zero vector by the transformation x Ax for the given matrix A. 5 -57 17. Let T: R → R? be a linear tran [3 and maps v = into [1 -3 that T is linear to find the image 3101
1. Let A = , and define T : R² → R² by T(x) = Ax. -6 10 %3D 2 -4 10. A = Find the images under T of u = 3 1 and v = 4. 10 8 3 2. Let A = a. u = and v = 11. Let b = and let A be the Define T : R in the range of the linear transforr why not? R' by T(x) = Ax. Find T(u) and T(v). %3D In Exercises 3-6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. %3D 12. Let b = and let A be the 0 -3 6 ,b = 1 -1 3. A = -3 1 3 b in the range of the linear transfor why not? 2 -2 -1 1 -2 1 -3 ,b = 3 4. A = In Exercises 13-16, use a rectangular -5 6. u = V = and their images 1 -5 mation T. (Make a separate and reason exercise.) Describe geometrically wha in R?. 5. A = -3 %3D 7 1 -3 27 8 ,b 13. T(x) = 3 -8 6. A = %3D 1 3 1 8. 10 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: R -→ R° by T(x) = Ax? 15. T(x) = 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R’ by the rule T(x) = Ax? 16. T) =: :: 16. T(x): For Exercises 9 and 10, find all x in R that are mapped into the zero vector by the transformation x Ax for the given matrix A. 5 -57 17. Let T: R → R? be a linear tran [3 and maps v = into [1 -3 that T is linear to find the image 3101
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Topic Video
Question
Number 10 show work
Transcribed Image Text:1. Suppose T : R$ → R² and T (x) = Ax for some matrix A and for each x in R’. How
many rows and columns does A have?
2. Let A =
Give a geometric description of the transformation x + Ax.
3. The line segment from 0 to a vector u is the set of points of the form tu, where
Ust<1. Show that a linear transformation T maps this segment into the segment
between 0 and T (u).
1.8 EXERCISES
1. Let A =
,and define T : R² → R² by T (x) = Ax.
3
2
10
-6
-4
1
10. A =
2
Find the images under T of u =
3
1
and v =
1
4
10
8
3
-1
2. Let A =
,u =
6.
and v = b
11. Let b =
and let A be the matrix in Exercise 9. Is E
Define T : R → R³ by T(x) = Ax. Find T(u) and T(v).
in the range of the linear transformation x + Ax? Why on
why not?
In Exercises 3–6, with T defined by T(x) = Ax, find a vector x
whose image under T is b, and determine whether x is unique.
3
12. Let b =
and let A be the matrix in Exercise 10. Is
1
0 -3
1
2 -2 -1
-1
3. А%3D
-3
b =
b in the range of the linear transformation x+ Ax? Why on
why not?
1 -2
1 -3
3
-6
4. A =
b =
-4
In Exercises 13-16, use a rectangular coordinate system to plo
2 -5
6
-5
u =
, and their images under the given transfor
5. A= =[
mation T. (Make a separate and reasonably large sketch for eacl
exercise.) Describe geometrically what T does to each vector >
in R2.
1
-3
1
-3
3 -8
8
,b =
13. Т(х) —
6. A =
1
1
8
14. T(x) =
7. Let A be a 6 x 5 matrix. What must a and b be in order to
define T: Rª → R³ by T(x) = Ax?
15. T(х) %3D
8. How many rows and columns must a matrix A have in order
to define a mapping from R$ into R' by the rule T(x) = Ax?
16. T(х) 3
For Exercises 9 and 10, find all x in R* that are mapped into the
zero vector by the transformation x Ax for the given matrix A.
17. Let T : R2 → R? be a linear transformation that maps u
into
and maps v =
into
Use the fa
-3
-5
that T is linear to find the images under T of 2u, 3v, ar
1 -3
2u + 3v.
9. A =
2 -4
4 -4
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