Explanation Given: The linear transformation T : R 4 → R 2 is defined as T ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 + x 2 + x 3 + x 4 , x 4 − x 1 ) , the point v= ( 4 , − 3 , 1 , 1 ) , B = { ( 1 , 0 , 0 , 1 ) , ( 0 , 1 , 0 , 1 ) , ( 1 , 0 , 1 , 0 ) , ( 1 , 1 , 0 , 0 ) } and B ′ = { ( 1 , 1 ) , ( 2 , 0 ) } Approach: For linear transformation T : R n → R m Here, 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 ] . The image of vector v = ( 5 , 4 ) can be calculated by multiplying the standard matrix A by the vector v as follows. T ( v ) = A v ... ( 1 ) The standard basis for R 4 is { e 1 , e 2 , e 3 , e 4 } . Here, e 1 = ( 1 , 0 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 , 0 ) , e 3 = ( 0 , 0 , 1 , 0 ) , and e 4 = ( 0 , 0 , 0 , 1 ) To determine (b) To find: The image of vector v = ( 4 , − 3 , 1 , 1 ) using the matrix relative to B and B ′

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

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ISBN: 9781337604925

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.3, Problem 40E

To determine

(a)

To find:

T(v) by using the standard matrix

To determine

(b)

To find:

The image of vector v=(4,−3,1,1) using the matrix relative to B and B′

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Chapter 6.3, Problem 39E

Chapter 6.3, Problem 41E

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Chapter 6 Solutions

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

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: T ( v ) by using the standard matrix

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

Videos

(a)

To find:

T(v) by using the standard matrix

(b)

To find:

The image of vector v=(4,−3,1,1) using the matrix relative to B and B′

Want to see the full answer?

Chapter 6 Solutions

(a) To find: T ( v ) by using the standard matrix

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

Videos

(a) To find: T(v) by using the standard matrix

(b) To find: The image of vector v=(4,−3,1,1) using the matrix relative to B and B′

Want to see the full answer?

Chapter 6 Solutions

(a)

To find:

T(v) by using the standard matrix

(b)

To find:

The image of vector v=(4,−3,1,1) using the matrix relative to B and B′