We say a linear transformation T : R
n → R
n
is norm preserving if kT(x)k = kxk, and inner-
product preserving if hT x, T yi = hx, yi.
(a) Prove that T is norm preserving if and only if T is inner-product preserving.
(b) Prove that such a linear transformation T is 1-1, invertible and T
−1
is also 1-1.

To address this problem, we will go step by step through each part.

### Part (a): Prove that \( T \) is norm preserving if and only if \( T \) is inner-product preserving.

#### 1. **Given**:
   - A linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \).
   - \( T \) is **norm-preserving** if \( \|T(x)\| = \|x\| \) for all \( x \in \mathbb{R}^n \).
   - \( T \) is **inner-product preserving** if \( \langle T(x), T(y) \rangle = \langle x, y \rangle \) for all \( x, y \in \mathbb{R}^n \).

#### 2. **Prove that norm-preserving implies inner-product preserving**:

Start with the definition of the norm in terms of the inner product:
\[
\|x\| = \sqrt{\langle x, x \rangle}
\]
Assume \( T \) is norm-preserving. This implies that for all \( x \in \mathbb{R}^n \):
\[
\|T(x)\| = \|x\|
\]
So, using the definition of the norm:
\[
\sqrt{\langle T(x), T(x) \rangle} = \sqrt{\langle x, x \rangle}
\]
Square both sides:
\[
\langle T(x), T(x) \rangle = \langle x, x \rangle \quad \text{(1)}
\]
Next, to show that \( T \) preserves the inner product, let's consider the inner product of two arbitrary vectors \( x \) and \( y \). Use the polarization identity for inner products:
\[
\langle x, y \rangle = \frac{1}{4} \left( \|x + y\|^2 - \|x - y\|^2 \right)
\]
Applying the same to \( T(x) \) and \( T(y) \):
\[
\langle T(x), T(y) \rangle = \frac{1}{4} \left( \|T(x) + T(y)\|^2 - \|T(x) - T(y)\|^2 \right)
\]
Since \( T \) is norm-preserving, \( \|T(x) + T(y)\| = \|x + y\| \) and \( \|T(x) - T(y)\| = \|x - y\| \), so:
\[
\langle T(x), T(y) \rangle = \frac{1}{4} \left( \|x + y\|^2 - \|x - y\|^2 \right) = \langle x, y \rangle
\]
This proves that \( T \) is inner-product preserving.

#### 3. **Prove that inner-product preserving implies norm-preserving**:

Assume \( T \) preserves the inner product, i.e., \( \langle T(x), T(y) \rangle = \langle x, y \rangle \) for all \( x, y \in \mathbb{R}^n \).

Now, set \( x = y \). Then:
\[
\langle T(x), T(x) \rangle = \langle x, x \rangle
\]
By the definition of the norm:
\[
\|T(x)\| = \sqrt{\langle T(x), T(x) \rangle} = \sqrt{\langle x, x \rangle} = \|x\|
\]
Thus, \( T \) is norm-preserving.

This completes the proof that \( T \) is norm-preserving if and only if \( T \) is inner-product preserving.

### Part (b): Prove that such a linear transformation \( T \) is 1-1, invertible, and \( T^{-1} \) is also 1-1.

#### 1. **Prove \( T \) is 1-1**:

To show that \( T \) is injective (1-1), suppose \( T(x) = T(y) \) for some \( x, y \in \mathbb{R}^n \). We need to show that \( x = y \).

Since \( T \) is inner-product preserving, we have:
\[
\langle T(x), T(y) \rangle = \langle x, y \rangle
\]
Given \( T(x) = T(y) \), it follows that:
\[
\langle T(x), T(y) \rangle = \langle T(x), T(x) \rangle = \langle x, x \rangle
\]
Thus, \( \langle x, y \rangle = \langle x, x \rangle \), which implies that \( x = y \).

Therefore, \( T \) is injective.

#### 2. **Prove \( T \) is invertible**:

Since \( T \) is a linear transformation from a finite-dimensional vector space \( \mathbb{R}^n \) to itself, and \( T \) is injective, by the rank-nullity theorem, \( T \) must also be surjective. Therefore, \( T \) is bijective and hence invertible.

#### 3. **Prove \( T^{-1} \) is also 1-1**:

Since \( T \) is invertible, \( T^{-1} \) exists. To show that \( T^{-1} \) is injective, suppose \( T^{-1}(x) = T^{-1}(y) \) for some \( x, y \in \mathbb{R}^n \). Apply \( T \) to both sides:
\[
T(T^{-1}(x)) = T(T^{-1}(y))
\]
This simplifies to:
\[
x = y
\]
Thus, \( T^{-1} \) is injective.

### Summary:

1. **(a)** We proved that a linear transformation \( T \) is norm-preserving if and only if it is inner-product preserving.
2. **(b)** We showed that such a linear transformation \( T \) is 1-1, invertible, and \( T^{-1} \) is also 1-1.

---

Would you like more details on any specific part, or do you have any questions?

Here are 5 related questions:
1. How does the rank-nullity theorem support the invertibility of a linear transformation?
2. What are the implications of a transformation being inner-product preserving in different geometric contexts?
3. Can a linear transformation be norm-preserving in only some dimensions but not others?
4. How does the concept of isometry relate to norm-preserving transformations?
5. How would you describe the geometric interpretation of a norm-preserving transformation?

**Tip:** To prove injectivity for transformations in finite dimensions, showing that the transformation has full rank (i.e., no kernel) is often a straightforward approach.

Explore the relationship between norm-preserving and inner-product preserving linear transformations. Understand the proofs and implications of these concepts through detailed examples and step-by-step explanations. Suitable for advanced undergraduate students studying linear algebra.

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