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Chapter 7: Q54E (page 375)

For Exercises 51 through 55, state whether the given set is a field (with the customary addition and multiplication).

54. The rotation-scaling matrices of the form [p-qqp], where p and q are real numbers

Short Answer

Expert verified

Yes, rotation-scaling matrices is a field

Step by step solution

01

Define rotation-scaling matrices

A rotation matrix and a translation matrix can be combined into a single matrix as follows, where the r's in the upper-left 3-by-3 matrix form a rotation and p, q and r form a translation vector.

02

Verify rotation-scaling matrices is field

Yes

The addition neutral is the matrix 0 , while the multiplication neutral is the matrix I2. Clearly, this set satisfies all the characteristic of a field, so this set is a field.

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