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Linear Algebra With Applications
Found in: Page 184
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, T(x+iy)=x2+y2 from to .

The transformation T(x+iy)=x2+y2 is not a linear transformation.

See the step by step solution

Step by Step Solution

Step 1: Definition of Linear Transformation

Consider two linear spaces V and W. A transformation T is said to be a linear transformation if it satisfies the properties,

T(f+g)=T(f)+T(g) T(kv)=kT(v)

For all elements f,g of v and k is scalar.

An invertible linear transformation is called an isomorphism.

Step 2: Check whether the given transformation is a linear or not.

Consider the transformation T(x+iy)=x2+y2 , from to .

Check whether the transformation satisfies the below two conditions or not.

1.T(A+B)=T(A)+T(B) 2.T(kA)=kT(A)

Verify the first condition.

Let A=x1+iy1 and B=x2+iy2 be arbitrary complex numbers from . Then,

T(A+B)=T(x1+iy1+x2+iy2) =Tx1+x2+iy1+y2 =x1+x22+y1+y22 =x12+x22+2x1x2+y12+y22+2y1y2 =x12+y12+x22+y22+2x1x2+2y1y2 =TA+TB+2x1x2+2y1y2

It is clear that, the first condition TA+BT(A)+T(B) is not satisfied.

Thus, T is not a linear transformation.

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