If there is a basis x1,...,xn of Rn and numbers λ1,...,λn such that Txi = λixi, prove that T is angle preserving if and only if all |λi| are equal.

Given a linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) and a basis \( \{x_1, \dots, x_n\} \) of \( \mathbb{R}^n \), suppose there exist scalars \( \lambda_1, \dots, \lambda_n \) such that

\[
T(x_i) = \lambda_i x_i \quad \text{for all } i = 1, \dots, n.
\]

We need to prove that \( T \) is angle-preserving if and only if \( |\lambda_1| = |\lambda_2| = \dots = |\lambda_n| \).

### Step 1: Understanding Angle Preservation

A linear transformation \( T \) is angle-preserving if for any two vectors \( u, v \in \mathbb{R}^n \), the angle between \( T(u) \) and \( T(v) \) is the same as the angle between \( u \) and \( v \). The angle \( \theta \) between two vectors \( u \) and \( v \) is given by:

\[
\cos \theta = \frac{\langle u, v \rangle}{\|u\| \|v\|}
\]

Therefore, \( T \) is angle-preserving if for any \( u, v \),

\[
\frac{\langle T(u), T(v) \rangle}{\|T(u)\| \|T(v)\|} = \frac{\langle u, v \rangle}{\|u\| \|v\|}
\]

### Step 2: Expressing the Condition Using the Given Basis

Consider the basis vectors \( x_i \) and \( x_j \). Under \( T \), we have:

\[
T(x_i) = \lambda_i x_i \quad \text{and} \quad T(x_j) = \lambda_j x_j
\]

We need to check if the angle between \( T(x_i) \) and \( T(x_j) \) is the same as the angle between \( x_i \) and \( x_j \). Since \( x_i \) and \( x_j \) form a basis, we can assume without loss of generality that \( \langle x_i, x_j \rangle = 0 \) for \( i \neq j \) (i.e., the basis is orthonormal). Therefore, the angle \( \theta_{ij} \) between \( x_i \) and \( x_j \) satisfies:

\[
\cos \theta_{ij} = \frac{\langle x_i, x_j \rangle}{\|x_i\| \|x_j\|} = 0
\]

For \( T \) to preserve angles, we require that

\[
\cos \theta_{ij}' = \frac{\langle T(x_i), T(x_j) \rangle}{\|T(x_i)\| \|T(x_j)\|} = 0
\]

Given the transformation \( T(x_i) = \lambda_i x_i \), the inner product \( \langle T(x_i), T(x_j) \rangle \) becomes:

\[
\langle T(x_i), T(x_j) \rangle = \langle \lambda_i x_i, \lambda_j x_j \rangle = \lambda_i \lambda_j \langle x_i, x_j \rangle = 0
\]

Thus, \( \cos \theta_{ij}' = 0 \), confirming that \( T \) does not affect orthogonality.

### Step 3: Norm Preservation and \( | \lambda_i | \) Condition

For \( T \) to be angle-preserving, not only must the inner product be preserved, but also the norms of the transformed vectors must scale proportionally. The norm of \( T(x_i) \) is:

\[
\|T(x_i)\| = |\lambda_i| \|x_i\|
\]

For angle preservation, the relative scaling between the norms of any two transformed basis vectors must be consistent, meaning:

\[
\frac{\|T(x_i)\|}{\|T(x_j)\|} = \frac{|\lambda_i| \|x_i\|}{|\lambda_j| \|x_j\|} = \frac{\|x_i\|}{\|x_j\|}
\]

This simplifies to:

\[
\frac{|\lambda_i|}{|\lambda_j|} = 1 \quad \text{for all } i, j
\]

Thus, \( |\lambda_i| = |\lambda_j| \) for all \( i, j \), implying that all \( |\lambda_i| \) are equal.

### Step 4: Conclusion

\( T \) is angle-preserving if and only if \( |\lambda_1| = |\lambda_2| = \dots = |\lambda_n| \). This completes the proof.

---

Do you want more details or have any questions? Here are five related questions to explore further:

1. How does orthogonality play a role in linear transformations preserving angles?
2. What would be an example of a linear transformation that is not angle-preserving?
3. Can angle preservation be extended to transformations on complex vector spaces?
4. What is the geometric interpretation of a linear transformation with unequal \( |\lambda_i| \)?
5. How does this proof relate to the concept of eigenvalues and eigenvectors?

**Tip:** When working with linear transformations, always consider the effects on both the length (norm) and direction of vectors.

Explore a proof that a linear transformation preserves angles if and only if all eigenvalues in its basis representation are of equal magnitude. This detailed mathematical analysis covers linear algebra, eigenvalues, and orthogonality.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation