Explanation Given: The linear transformation is as follows, T ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 − x 2 , x 3 , x 1 + 2 x 2 − x 4 , x 4 ) Approach: The standard basis for R 3 is as follows, B 1 = { e 1 = [ 1 0 0 ] , e 2 = [ 0 1 0 ] , e 3 = [ 0 0 1 ] } . If T : R n → R m such that T ( e 1 ) = [ a 11 a 21 ⋮ a m 1 ] , ⋯ , T ( e n ) = [ a 1 n a 2 n ⋮ a m n ] Then 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 ] . Calculation: Let the standard basis B be B = { e 1 = ( 1 , 0 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 , 0 ) , e 3 = ( 0 , 0 , 1 , 0 ) , e 4 = ( 0 , 0 , 0 , 1 ) } . Then, T ( e 1 ) = ( 1 − 0 , 0 , 1 + 2 ( 0 ) − 0 , 0 ) = ( 1 , 0 , 1 , 0 ) Thus, T ( e 1 ) = [ 1 0 1 0 ] Again, To determine (b) To find: The image of the given vector v = ( 1 , 0 , 1 , − 1 ) .

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 25E

To determine

(a)

To find:

The standard matrix A for T(x1,x2,x3,x4)=(x1−x2,x3,x1+2x2−x4,x4).

To determine

(b)

To find:

The image of the given vector v=(1,0,1,−1).

Expert Solution & Answer

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 standard matrix A for T ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 − x 2 , x 3 , x 1 + 2 x 2 − x 4 , x 4 ) .

Solution Summary: The author explains that the standard matrix is A for T(x_1,X2 and R3.

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(a)

To find:

The standard matrix A for T(x1,x2,x3,x4)=(x1−x2,x3,x1+2x2−x4,x4).

(b)

To find:

The image of the given vector v=(1,0,1,−1).

Chapter 6 Solutions

(a) To find: The standard matrix A for T ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 − x 2 , x 3 , x 1 + 2 x 2 − x 4 , x 4 ) .

Solution Summary: The author explains that the standard matrix is A for T(x_1,X­2 and R3.

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(a) To find: The standard matrix A for T(x1,x2,x3,x4)=(x1−x2,x3,x1+2x2−x4,x4).

(b) To find: The image of the given vector v=(1,0,1,−1).

Chapter 6 Solutions

Solution Summary: The author explains that the standard matrix is A for T(x_1,X2 and R3.

(a)

To find:

The standard matrix A for T(x1,x2,x3,x4)=(x1−x2,x3,x1+2x2−x4,x4).

(b)

To find:

The image of the given vector v=(1,0,1,−1).