Q7E Question 7: Suppose v1鈫�,v2鈫�,... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q7E (page 53)

Question 7: Suppose $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}} 鈰� \overset{鈫�}{v_{m}}$ are arbitrary vectors in Consider the linear transformation from togiven by
$T [x_{1} x_{2} 鈰� x_{m}] = [x_{1} \overset{鈫�}{v_{1}}, x_{2} \overset{鈫�}{v_{2}}, x_{3} \overset{鈫�}{v_{3}} 鈰� x_{m} \overset{鈫�}{v_{m}}]$

Short Answer

Expert verified

Answer:

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

Step by step solution

Step1: System of transformation

We have given a transformation with $T [x_{1} x_{2} 鈰� x_{m}] = [x_{1} \overset{鈫�}{v_{1}}, x_{2} \overset{鈫�}{v_{2}}, x_{3} \overset{鈫�}{v_{3}} 鈰� x_{m} \overset{鈫�}{v_{m}}]$ .

Linear Transformation

A transformation from $R^{m} 鈫� R^{n}$ 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 $a, b 鈭� R^{m}$ .
$T (c a) = c T (a)$ Where c is any scalar and $a 鈭� R^{m}$ .

Checking for linear transformation

Now we will find the transformation of identity element.

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

Now let $a, b 鈭� R^{m}$ .

Then the transformation

$T (伪 a + 尾 b) = 伪 T (a) + 尾 T (b)$

Thus, all the conditions are true.

Matrix for linear transformation

The matrix represented by. $A = [\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}} 鈰� \overset{鈫�}{v_{m}}]$

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

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91影视

Short Answer

Step by step solution

Step1: System of transformation

Linear Transformation

Checking for linear transformation

Matrix for linear transformation

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91影视

Question 7: Suppose v1鈫�,v2鈫�,v3鈫�鈰�vm鈫�are arbitrary vectors in Consider the linear transformation from togiven by T[x1x2鈰�xm]=[x1v1鈫�,x2v2鈫�,x3v3鈫�鈰�xmvm鈫�]

Short Answer

Step by step solution

Step1: System of transformation

Linear Transformation

Checking for linear transformation

Matrix for linear transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Logic and Functions

Discrete Mathematics

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.