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Chapter 5: Q 5.4-14E (page 246)

Question: In the accompanying figure, we show the kernel and the image of a linear transformation Lfrom R2 to R2 , together with some vectors. v鈫�1,w鈫�1,w鈫�2,w鈫�3We are told that L(v鈫�1)=w鈫�1. For i = 1,2,3 find the vectors L+(w鈫�i), where L+s the pseudoinverse of Ldefined in Exercise 13. Show your solutions in the figure, and explain how you found them.

Short Answer

Expert verified

L+(w鈫�2)=L+(projim(L)w鈫�2)=L+(0鈫�)=0鈫�L+(w鈫�3)=L+(projim(L)w鈫�3)

Step by step solution

01

Minimal solution of Ker (L).

Since L+(w鈫�1)is the minimal solution of the system . Thus, from Exercise 9, the minimal solution is perpendicular to Ker (L).

Consider the solution below

L+(w鈫�2)=L+(projim(L)w鈫�2)=L+(0鈫�)=0鈫�L+(w鈫�3)=L+(projim(L)w鈫�3)

Consider the diagram below.L+x鈫�=w鈫�

Hence,

L+(w鈫�2)=L+(projim(L)w鈫�2)=L+(0鈫�)=0鈫�L+(w鈫�3)=L+(projim(L)w鈫�3)inthediagram.

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Most popular questions from this chapter

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?AB.

Consider a linear transformationL from RmtoRn that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If Ais the matrix of L, What can you say about the columns of A? What isATA? What aboutAAT? Illustrate your answer with an example where m=2and n=3.

If A is an n脳nmatrix such that role="math" localid="1659514225617" AAT=In, then A must be an orthogonal matrix.

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? 17[26-36-32326].

In Exercises 40 through 46, consider vectors in ; we are told that is the entry of matrix A.

A=[35115920112049]

46. Find , where V =span . Express your answer as a linear combination of and .
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