Suppose that B = S − 1 A S for some n × n matrices A , B , and S . a. Show that if x → is in kerB , then S x → is in kerA . b. Show that the linear transformation T ( x → ) = S x → from kerB to kerA is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B .
Suppose that B = S − 1 A S for some n × n matrices A , B , and S . a. Show that if x → is in kerB , then S x → is in kerA . b. Show that the linear transformation T ( x → ) = S x → from kerB to kerA is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B .
Solution Summary: The author explains that the linear transformation T(stackrelto x)=S
Suppose that
B
=
S
−
1
A
S
for some
n
×
n
matrices A, B, and S. a. Show that if
x
→
is in kerB, then
S
x
→
is in kerA. b. Show that the linear transformation
T
(
x
→
)
=
S
x
→
from kerB to kerA is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B.
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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