5 Most Amazing To Inversion Theorem
Proof:14 First assume
X
{\displaystyle X}
is compact. Let us understand the concept of inverse of algebraic relation with the help of an example,Consider a relation in its algebraic formR = {(x,y) : y = 2x + 4}We find the inverse of the given algebraic expression from the following steps,We interchange x and y,So the new algebraic expression becomes,x = 2y + 4x 4 = 2y(x-4) / 2 = yHence the inverse of the given algebraic expression shall be,R⁻¹ = {(x,y) : y = (x-4)/2}Hence, we get, y = 2x + 4 and y = (x-4) / 2 and will be symmetric about the line y = x. Using the geometric series for
B
=
I
{\displaystyle B=I-A}
, it follows that
A
1
2
{\displaystyle \|A^{-1}\|2}
.
When
f
:
R
n
R
m
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}
with
m
n
{\displaystyle m\leq n}
,
f
{\displaystyle f}
is
k
{\displaystyle k}
times continuously differentiable, and the Jacobian
A
=
straight from the source
f
(
x
)
{\displaystyle A=\nabla f({\overline {x}})}
at a point
x
{\displaystyle {\overline {x}}}
is of rank
m
{\displaystyle m}
, the inverse of
f
{\displaystyle f}
may not be unique.
5 Steps to Commonly Used Designs
.