Could you explain how to show this in detail? Consider a matrix A of the formA- (: )-adwith a, b, c, d ER not all equal to zero. Define the linear map fa induced by A by()() - (: )() - )(u(x, y)\v(x, y),b.ах + byсх + dyafa:E R HE R? .d.(i) Compute the Jacobi matrix DfA(x,y) at (x, y) E R².(ii) Show that ƒa is real differentiable at every (x, y) E R².(iii) Write z = x + iy, w = u + iv, and regard fA as a function ƒA: C → C, namelyw = fA(2) = Az. Show that fa is complex differentiable if and only ifd = aandc = -b.(1)(iv) Show that if (1) holds then there are p> 0 and a unique a E (-1, T] such thatCOs a– sin aA = psin aCOS a(v) Show that if (1) holds then there exists a number A E C such that faA(2) = \z, namely,that fa is the rule that associates to every z the result of its multiplication times A:this is called the multiplication operator with factor A.

Name: Could you explain how to show this in detail? Consider a matrix A of t
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Could you explain how to show this in detail? Consider a matrix A of the formA- (: )-adwith a, b, c, d ER not all equal to zero. Define the linear map fa induced by A by()() - (: )() - )(u(x, y)\v(x, y),b.ах + byсх + dyafa:E R HE R? .d.(i) Compute th

Answered: Consider a matrix A of the form A- (:)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Topic Video

Question

100%

Could you explain how to show this in detail?

Consider a matrix A of the form
A- (: )-
a
d
with a, b, c, d ER not all equal to zero. Define the linear map fa induced by A by
()
() - (: )() - )
(u(x, y)\
v(x, y),
b.
ах + by
сх + dy
a
fa:
E R H
E R? .
d.
(i) Compute the Jacobi matrix DfA(x,y) at (x, y) E R².
(ii) Show that ƒa is real differentiable at every (x, y) E R².
(iii) Write z = x + iy, w = u + iv, and regard fA as a function ƒA: C → C, namely
w = fA(2) = Az. Show that fa is complex differentiable if and only if
d = a
and
c = -b.
(1)
(iv) Show that if (1) holds then there are p> 0 and a unique a E (-1, T] such that
COs a
– sin a
A = p
sin a
COS a
(v) Show that if (1) holds then there exists a number A E C such that faA(2) = \z, namely,
that fa is the rule that associates to every z the result of its multiplication times A:
this is called the multiplication operator with factor A.

Expert Solution

Step 1

As per bartleby guidelines for more than three subparts in a question only first three should be answered. Please upload others separately.

(i) Given that $f_{A} (x, y) = (a x + b y, c x + d y)$

The Jacobian matrix , $D f (x, y) = [\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{matrix}]$

Here $u = a x + b y a n d v = c x + d y$

Therefore,

$D f_{A} (x, y) = [\begin{matrix} a & b \\ c & d \end{matrix}]$

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Propositional Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions