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Chapter 3: Q27E (page 120)

Give an example of a noninvertible function Ffrom鈩�to 鈩浓划with

im(f)=鈩�

Short Answer

Expert verified

fx=x3-x is a noninvertible function f from鈩浓划 to鈩� withimf=鈩� .

Step by step solution

01

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

imagef=fx:xinX=binY:b=fx,forsomexinX

The image of the function is,

F:imagef鈫�power鈩�鈬�f:鈩�鈫�P鈩�

For example, fx=x3-xis an example of a noninvertible function f from 鈩浓划 to 鈩�with localid="1660366374594" imf=鈩�imf=鈩�.

02

Final answer.

fx=x3-xisanoninvertiblefunctionffrom鈩浓划鈩�to鈩浓划鈩�withim(f)=R.

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Most popular questions from this chapter

Question: Consider linearly independent vectors v1鈫�,v2鈫�,....,vm鈫�in 鈩�nand let A be an invertible m脳mmatrix. Are the columns of the following matrix linearly independent?

[IV1IV2...IVmIII]A

Find a basis of the kernel of the matrix

[1203500146]

Justify your answer carefully; that is, explain how you know that the vectors you found are linearly independent and span the kernel.

Give an example of a matrixAsuch thatim(A)is the plane with normal vector [132]in鈩�3鈩�3 .

Explain why you need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. See Theorem 3.3.4b.

In Exercises 37 through 42 , find a basis I of 鈩�n such that the I-matrix of the given linear transformation T is diagonal.

Orthogonal projection T onto the plane 3x1+x2+2x3=0 in鈩�3.
See all solutions

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