i need this question completed in 5 minutes with handwritten working out 8.(a) Define the kernel of a linear operator L: U → V.(b) Define the adjoint of a linear operator L: U → V.(c) Find the adjoint of the matrixA=(13)with the following inner products:(i) the Euclidean dot product on R², for both the domain and the targetspace, and(ii) the weighted inner product (v, w) = 2v1w₁ +3v2w2 on R², for both thedomain and target space.

Given Information:The matrix is provided as .The inner product is of Euclidean dot product and the…

Answered: (a) Define the kernel of a linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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i need this question completed in 5 minutes with handwritten working out

8.
(a) Define the kernel of a linear operator L: U → V.
(b) Define the adjoint of a linear operator L: U → V.
(c) Find the adjoint of the matrix
A
=
(13)
with the following inner products:
(i) the Euclidean dot product on R², for both the domain and the target
space, and
(ii) the weighted inner product (v, w) = 2v1w₁ +3v2w2 on R², for both the
domain and target space.

Expert Solution

Step 1: Analysis and Introduction

Given Information:

The matrix is provided as $A equals open parentheses table row 1 2 row cell negative 1 end cell 3 end table close parentheses$ .

The inner product is of Euclidean dot product and the weighted inner product is defined as $open angle brackets v comma w close angle brackets equals 2 v subscript 1 w subscript 1 plus 3 v subscript 2 w subscript 2$ .

Objective:

a) Define the Kernel of Linear operator $L colon U rightwards arrow V$ .

b) Define the Adjoint of Linear operator $L colon U rightwards arrow V$ .

c) Find the Adjoint of the matrix on Euclidean dot product and weighted inner product.

Concept used:

Suppose $A$ is the matrix of linear transformation $T colon V rightwards arrow W$ .

The adjoint of the matrix $A asterisk times$ with the property that $open angle brackets A x comma y close angle brackets equals open angle brackets x comma A asterisk times y close angle brackets$ .

The Euclidean dot product of two vectors $v equals open parentheses v subscript 1 comma v subscript 2 close parentheses comma space w equals open parentheses w subscript 1 comma w subscript 2 close parentheses$ is $v times w equals v subscript 1 w subscript 1 plus v subscript 2 w subscript 2$ .