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Chapter 4: Q2E (page 184)

Question: Find the transformation is linear and determine whether they are isomorphism.

Short Answer

Expert verified

The solution is linear and isomorphism.

Step by step solution

01

Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

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

An invertible linear transformation is called an isomorphism.

02

Explanation of the solution

The given transformation as follows.

TM=7Mfrom R2脳2toR2脳2.

By using the definition of linear transformation as follows.

T(A+B)=7(A+B)=7A+7BT(A+B)=TA+TB

Similarly for scalars as follows.

TkA=7kA=k7ATkA=kTA

This shows that T is a linear transformation.

Now, to show that T is isomorphism.

That is to solve that T (A) = B for A.

TA=B7A=BA=17B

This, shows that the inverse transformation isTB=17B and T is isomorphism.

Thus, T is a linear transformation and isomorphism.

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