For Problems 10 − 14 , find Ker ( T ) and Rng ( T ) , and hence, determine whether the given transformation is one-to-one, onto, both, or neither. If T − 1 exists, find it. T ( x ) = A x , where A = [ 4 2 1 3 ] .
For Problems 10 − 14 , find Ker ( T ) and Rng ( T ) , and hence, determine whether the given transformation is one-to-one, onto, both, or neither. If T − 1 exists, find it. T ( x ) = A x , where A = [ 4 2 1 3 ] .
Solution Summary: The author explains the inverse of a transformation exists only when the transformation is both one-to-one and onto.
For Problems
10
−
14
, find
Ker
(
T
)
and
Rng
(
T
)
, and hence, determine whether the given transformation is one-to-one, onto, both, or neither. If
T
−
1
exists, find it.
For Problems 34-37, determine whether the linear transformation T is invertible. Either formally prove that your
answer is correct, or justify your answer by invoking appropriate definitions and theorems to explain the significance
of any calculations you do. Clearly state your final answer.
a + b
а — b
T:R? → M2x2 (R) defined by T(a, b) = ( |
2a – b 2a +b
Determine whether the following are linear transformationsfrom R3 into R2. L (x) = (x3, x1 + x2)T
5. Describe each of the following
transformations on y = x2.
%3D
а. у%3D 2x2+ 3
b. у %3D (Зx + 7)?
c. y = -0.5(x + 2)2 + 4
d. y = -(x - 1)2 - 2
Chapter 6 Solutions
Differential Equations and Linear Algebra (4th Edition)
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