Explanation Given: The linear transformation T : R 3 → R 3 is defined as T ( x , y , z ) = ( x + y + z , 2 z − x , 2 y − z ) , the vector is v = ( 4 , − 5 , 10 ) , the bases are B = { ( 2 , 0 , 1 ) , ( 0 , 2 , 1 ) , ( 1 , 2 , 1 ) } and B ′ = { ( 1 , 1 , 1 ) , ( 1 , 1 , 0 ) , ( 0 , 1 , 1 ) } . Approach: For linear transformation T : R n → R m where T ( e 1 ) = [ a 11 a 21 ⋮ a m 1 ] , T ( e 2 ) = [ a 12 a 22 ⋮ a m 2 ] , ⋯ T ( e n ) = [ a 1 n a 2 n ⋮ a m n ] , the standard matrix is formed as A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] . Here, e 1 , e 2 , ... , e n are standard basis of R n . The image of vector v = ( 4 , − 5 , 10 ) can be calculated by multiplying the standard matrix A by the vector v as, T ( v ) = A v Calculation: Calculate T ( e 1 ) , T ( e 2 ) and T ( e 3 ) To determine (b) To find: The image of vector v = ( 4 , − 5 , 10 ) using the matrix relative to B and B ′ .

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + MindTap Math, 1 term (6 months) Printed Access Card

8th Edition

ISBN: 9781337131216

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.3, Problem 41E

To determine

(a)

To find:

The value of T(v) by using the standard matrix for v=(4,−5,10).

To determine

(b)

To find:

The image of vector v=(4,−5,10) using the matrix relative to B and B′.

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Chapter 6.3, Problem 40E

Chapter 6.3, Problem 42E

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Chapter 6 Solutions

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + MindTap Math, 1 term (6 months) Printed Access Card

Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Prob. 4E Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Finding an Image and a PreimageIn Exercises 1-8,...Ch. 6.1 - Linear TransformationsIn Exercises 9-22, determine...Ch. 6.1 - Linear TransformationsIn Exercises 9-22, determine...

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(a) To find: The value of T ( v ) by using the standard matrix for v = ( 4 , − 5 , 10 ) .

Solution Summary: The author explains how to find the value of T(v) by using the standard matrix.

Videos

(a)

To find:

The value of T(v) by using the standard matrix for v=(4,−5,10).

(b)

To find:

The image of vector v=(4,−5,10) using the matrix relative to B and B′.

Want to see the full answer?

Chapter 6 Solutions

(a) To find: The value of T ( v ) by using the standard matrix for v = ( 4 , − 5 , 10 ) .

Solution Summary: The author explains how to find the value of T(v) by using the standard matrix.

Videos

(a) To find: The value of T(v) by using the standard matrix for v=(4,−5,10).

(b) To find: The image of vector v=(4,−5,10) using the matrix relative to B and B′.

Want to see the full answer?

Chapter 6 Solutions

(a)

To find:

The value of T(v) by using the standard matrix for v=(4,−5,10).

(b)

To find:

The image of vector v=(4,−5,10) using the matrix relative to B and B′.