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Chapter 4: Q5E (page 184)

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

Short Answer

Expert verified

The solution is not linear.

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.

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

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.

T(M)=M2fromR2脳2 to R2脳2.

By using the definition of linear transformation as follows.

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

Consider the2脳2 matrix forl2 as follows.

l2=10020

Substitute the value 2 for k inT2l2 and simplify as follows.

T2l2=2l22=4l2

Similarly, simplify for 2T( A ) as follows.

2TA=2l22=2l2

Since,T(2l2)鈮�2T(l2) and T is not linear.

Hence, the transformation T is not linear.

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Most popular questions from this chapter

Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets ofP2 given in Exercises 1through 5are subspaces ofrole="math" localid="1659355918761" P2 (see Example)? Find a basis for those that are subspaces,{p(t):p(2)=0}.

Find the basis of each of the space V=Rm脳n, and determine its dimension.

Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V ? The arithmetic sequences [i.e., sequences of the form(a,a+k,a+2k,a+3k,...) , for some constants and K .

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 鈩�to 鈩�.

Show that if Tis a linear transformation from Vto W, thenT(0v)=0wwhererole="math" localid="1659425903549" 0vand0ware the neutral elements of Vand W, respectively. If T is an isomorphism, show that T-1(0w)=0v.
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