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Linear Algebra With Applications
Found in: Page 185
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

For which constant k is a linear transformation T(M)=[2304]M-M[300k]is an isomorphism form R2×2 to R2×2.

The solution is an isomorphism when {kR-{2,4}.

See the step by step solution

Step by Step Solution

Step1: Definition of Linear Transformation

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

T(f+g)=T(f)+T(g) T(kf)=kT(f)

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

A linear transformation T:VW is said to be an isomorphism if and only if ker(T)={0} and im(T)=W or dim(V)=dim(W).

Step2: Explanation of the solution

The given transformation as follows.

TM=2304M-M300k, from R2×2 to R2×2.

By using the definition of linear transformation as follows.


Now, to check the first condition as follows.

Let A and B be arbitrary matrices from R2×2 and as follows.

TA+B=2304A+BA-B300k =2304A+2304B-A300k-B300k =230kA-A300k+2304B-B300kTA+B=TA+TB

Similarly, to check the second condition as follows.

Let α be an arbitrary scalar, and AR2×2as follows.

TαA=2304αA-αA300k =α2304A-A300kTαA=αTA

Thus,T is a linear transformation.

Step3: Properties of isomorphism

A linear transformation T:VW is isomorphism if and only if ker(t)={0}and localid="1659426664071" Im(t)=W

Now, check if ker(t)=0 as follows.


Consider a matrix A as follows.


The next equation as follows.

TA=0000 2304abcd-abcd300k=00002a+3c2b+3d0a+4c0d+4d-3a+0b0a+kb3c+0d0c+kd=00002a+3c-3a-0d0a+4c-3c-3d2b+3d-0a-kd0b+4d-0c-kd=0000

Simplify further as follows.


Equating the corresponding entries as follows.

c=0 and a+3c=0

Solve and find the values as follows.


Substitute the value 0 for and 0 for in A=abcdas follows.

A=abcd =0b0d

For k=4 the solution as follows.

T0001=0000 kerT0000 kerT0

Therefore,T to be an isomorphism for k4.

Similarly, for T to be an isomorphism for k2.

Thus, is a linear transformation and is an isomorphism when kR-2,4

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