Guided Proof Let B be an invertible n × n matrix. Prove that the linear transformation T : M n , n → M n , n represented by T ( A ) = A B is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8 , you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is { 0 } (Theorem 6.6 ). Use the fact that B is an invertible n × n matrix and that T ( A ) = A B . (iii) Conclude that T is an isomorphism.
Guided Proof Let B be an invertible n × n matrix. Prove that the linear transformation T : M n , n → M n , n represented by T ( A ) = A B is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8 , you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is { 0 } (Theorem 6.6 ). Use the fact that B is an invertible n × n matrix and that T ( A ) = A B . (iii) Conclude that T is an isomorphism.
Solution Summary: The author explains that the linear transformation T:M_n,nto
Guided Proof Let
B
be an invertible
n
×
n
matrix. Prove that the linear transformation
T
:
M
n
,
n
→
M
n
,
n
represented by
T
(
A
)
=
A
B
is an isomorphism.
Getting started: To show that the linear transformation is an isomorphism, you need to show that
T
is both onto and one-to-one.
(i)
T
is a linear transformation with vector spaces of equal dimension, so by Theorem
6.8
, you only need to show that
T
is one-to-one.
(ii) To show that
T
is one-to-one, you need to determine the kernel of
T
and show that it is
{
0
}
(Theorem
6.6
). Use the fact that
B
is an invertible
n
×
n
matrix and that
T
(
A
)
=
A
B
.
(iii) Conclude that
T
is an isomorphism.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY