Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Question 7: Suppose $\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}$ are arbitrary vectors in Consider the linear transformation from to given by
$T [x_{1} x_{2} ⋮ x_{m}] = [x_{1} \vec{v_{1}}, x_{2} \vec{v_{2}}, x_{3} \vec{v_{3}} \dots x_{m} \vec{v_{m}}]$

Answer:

Yes, given transformation is linear and the matrix represented by $A = [\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}]$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: System of transformation

We have given a transformation with $T [x_{1} x_{2} ⋮ x_{m}] = [x_{1} \vec{v_{1}}, x_{2} \vec{v_{2}}, x_{3} \vec{v_{3}} \dots x_{m} \vec{v_{m}}]$ .

Step2: Linear Transformation

A transformation from $R^{m} \to R^{n}$ is said to be linear if the following condition holds:

Identity of R^m should be mapped to identity of Rⁿ.
$T (a + b) = T (a) + T (b)$ For all $a, b \in R^{m}$ .
$T (c a) = c T (a)$ Where c is any scalar and $a \in R^{m}$ .

Step3: Checking for linear transformation

Now we will find the transformation of identity element.

$T (0, 0 . . . 0) = (0, 0 . . . 0)$

Now let $a, b \in R^{m}$ .

Then the transformation

$T (α a + β b) = α T (a) + β T (b)$

Thus, all the conditions are true.

Step4: Matrix for linear transformation

The matrix represented by. $A = [\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}]$

Hence, yes the given transformation is linear and the matrix represented by $A = [\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}]$ .

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Linear Algebra With Applications

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Short Answer

Question 7: Suppose $\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}$ are arbitrary vectors in Consider the linear transformation from to given by
$T [x_{1} x_{2} ⋮ x_{m}] = [x_{1} \vec{v_{1}}, x_{2} \vec{v_{2}}, x_{3} \vec{v_{3}} \dots x_{m} \vec{v_{m}}]$

Step by Step Solution

Step1: System of transformation

Step2: Linear Transformation

Step3: Checking for linear transformation

Step4: Matrix for linear transformation

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Linear Algebra With Applications

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Short Answer

Question 7: Suppose v1→,v2→,v3→⋯vm→are arbitrary vectors in Consider the linear transformation from to given by T[x1x2⋮xm] =[x1v1→,x2v2→,x3v3→⋯xmvm→]

Step by Step Solution

Step1: System of transformation

Step2: Linear Transformation

Step3: Checking for linear transformation

Step4: Matrix for linear transformation

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Question 7: Suppose $\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}} \dots \vec{v_{m}}$ are arbitrary vectors in Consider the linear transformation from to given by
$T [x_{1} x_{2} ⋮ x_{m}] = [x_{1} \vec{v_{1}}, x_{2} \vec{v_{2}}, x_{3} \vec{v_{3}} \dots x_{m} \vec{v_{m}}]$