be a linear transformation that preserves lengths; that

Chapter , Problem 8E

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QUESTION:

Problem 8E

be a linear transformation that preserves lengths; that is,

.

a. Show that T also preserves orthogonality; that is,

b. Show that the standard matrix of T is an orthogonal matrix.

Questions & Answers

QUESTION:

Problem 8E

be a linear transformation that preserves lengths; that is,

.

a. Show that T also preserves orthogonality; that is,

b. Show that the standard matrix of T is an orthogonal matrix.

ANSWER:

Solution 8E

Step 1 of 3

a)

Suppose  are two vectors such that

It is stated that the linear transformation, preserves lengths

That is  for all  in  .

The strategy of solution is to show that  preserves the orthogonality.

The norm of sum of vectors is,

 Since

By the Pythagorean Theorem we have

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