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Chapter 2: Q35E (page 73)

Let Tbe an invertible linear transformation from R2to R2. Let Pbe a parallelogram inR2with one vertex at the origin. Is the image of Pa parallelogram as well? Explain. Draw a sketch of the image.

Short Answer

Expert verified

Yes, the image of P is also a parallelogram as well. The required sketch is,

Step by step solution

01

Consider the linear transformation

The vector formation of the given parallelogram is,

p鈫�=ap1鈫�+bp2鈫�

The linear transformation is,

T(p鈫�)=T(ap1鈫�+bp2鈫�)

02

Compute the condition.

Consider the linear transformation is,

T(p鈫�)=T(ap1鈫�+bp2鈫�)

Use the properties of the linear transformation in the above equation.

T(p鈫�)=T(ap1鈫�)+T(bp2鈫�)鈬�T(p鈫�)=a(Tp1鈫�)+b(Tp2鈫�)

This represents that the image of Pis also a parallelogram as well.

Hence, the image ofP is also a parallelogram as well.

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Most popular questions from this chapter

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix [0.40.30.60.7]andx炉is the distribution vector[10]. [We will not be using parts (a) and (c) of Theorem 2.3.11]. The general proof of Theorem 2.3.11 runs along similar lines, as we will see in Chapter 7.

  1. ComputeA[12]andA[1-1]. WriteA[1-1]as a scalar multiple of the vector[1-1].
  2. Write the distribution vectorx鈫�=[10]as a linear combination of the vectors[12]and[1-1]
  3. Use your answers in part (a) and (b) to writeAx鈫�as a linear combination of the vectors[12]and[1-1]. More generally, write Amx鈫�as a linear combination of vectors[12]and[1-1], for any positive integer m. See Exercise 81.
  4. In your equation in part (c), let got to infinity to find limm鈫�鈭�Amx鈫�. Verify that your answer is the equilibrium distribution for A.

Question 7: Suppose v1鈫�,v2鈫�,v3鈫�鈰�vm鈫�are arbitrary vectors in Consider the linear transformation from togiven by

T[x1x2鈰�xm]=[x1v1鈫�,x2v2鈫�,x3v3鈫�鈰�xmvm鈫�]

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?24 f(x)=x3-x.

TRUE OR FALSE?

There exists an upper triangular2脳2 matrixAsuch that A2=(1101) .

Which of the functions f from R toR in Exercises 21 through 24 are invertible?22 f(x)=2x.
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