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Chapter 4: Q43E (page 200)

To determine whether the given statement 鈥淭hat the inequalitydim(W)鈮�5 holds if T is a linear transformation from P4to W withIm(T)=W. .

Short Answer

Expert verified

The solution is the statement is true.

Step by step solution

01

Explanation of the solution

Consider that T is a linear transformation fromP4 to W with Im(T)=W..

By the definition as follows.

Im(T))={T(f)):f鈭�P4}lm(T)=W

02

Find the dimension of W

This implies that follows:

dim(W)鈮�dim(P4)dim((W)鈮�5

Therefore, the inequalitydim(W)鈮�5 holds if T is a linear transformation from role="math" localid="1659434597281" P4to W with Im(T)=W.

Thus, the given statement 鈥渢he inequalitydim(W)鈮�5 holds if T is a linear transformation from P4to W with Im(T)=W鈥� is true.

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Most popular questions from this chapter

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�.

Find the basis of all2脳2 matrixA such thatA=[1101]=[0000], and determine its dimension.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,T(ft)=鈭�-23f(t)dt from toP2to鈩�.

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 of given in Exercises 1through 5are subspaces ofP2 (see Example 16)? Find a basis for those that are subspaces,{p(t):p(0)=2}.

T denotes the space of infinity sequence of real numbers,T(f(t))=[f(5)f(7)f(11)]fromP2toR3. .
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