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Chapter 4: Q24E (page 200)

State true or false, if S is any invertible 2脳2 matrix, then the linear transformation T(M)=SMS is an isomorphism fromR2脳2toR2脳2

Short Answer

Expert verified

The given statement is True.

Step by step solution

01

Determine the isomorphism of T.

Consider a linear transformation T from R2脳2toR2脳2as T(M)=SMS where S is an invertible matrix.

If the matrix Sis invertible thenS鈮�0.

Theorem: Consider a linear transformation T defined from T:V 鈫�Wthen the transformation Tis an isomorphism if dim(ker(T))=0 and only if where

ker(T)={fxI^P:TFx=0}anddim(kerT)={fxI^P:Tfx=0impliesfx=0}

A kernel of a function Tis defined as ker(T)={fxI^P:Tfx=0}.

02

Determine if the statement is True or false

IfSMS=0impliesM=0meansT(M)=0impliesM=0

By the definition of kernel, the dimension of kernel of T is0 .

By the theorem, the linear transformation T is isomorphism.

Hence, the statement is true.

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