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Chapter 6: Q20E (page 307)

Question:We say that a linear transformation Tfrom N3toN3to preserves orientation if it transforms any positively oriented basis into another positively oriented basis. See Exercise 19. Explain why a linear transformationT(x鈫�)=Ax鈫�preserves orientation if (and only if) detAis positive.

Short Answer

Expert verified

Therefore, the linear transformationT:N3鈫�N3,T(x鈫�)=Ax鈫� whereA鈭�N3脳3 ,preserves orientation if and only if detTB=detTdetB>0, which is if and only ifdetT=detA>0

Step by step solution

01

Definition.

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

Explanation why a linear transformation T(x→)=Ax→ preserves orientation if (and only if) detA is positive.

For an arbitrary positively oriented basis B=v1鈫�,v2鈫�,v3鈫�forN3, letB=v1鈫�v2鈫�v3鈫�

We have B > 0.

A linear transformation T:N3鈫�N3,T(x鈫�)=Ax鈫�, where A鈭�N3脳3, preserves orientation if and only if detTB=detTdetB>0, which is if and only if detT=detA>0.

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Most popular questions from this chapter

In an economics text11we find the following system:

[-R1R1-1-伪伪1-伪-1-伪2R2-R2-1-伪2伪][dx1dy1dp]=[00-R2de2].

Solve for dx1,dy1, and dp. In your answer, you may refer to the determinant of the coefficient matrix as D. (You need not compute D.) The quantitiesR1,R2and D are positive, and ais between zero and one. If de2is positive, what can you say about the signs of dy1and dp?

If all the diagonal entries of an n脳nmatrixA are even integers and all the other entries are odd integers, then Amust be an invertible matrix.

Consider a4脳4matrix A with rowsv鈫�1,v鈫�2,v鈫�3,v鈫�4. If det(A) = 8, find the determinants in Exercises 11 through16.

13. det[v鈫�1v鈫�2v鈫�3v鈫�4]

det[0100001000011000]=1

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

8. [0000210003010040010500016]
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