What is the definition of a linear isometry between two inner product spaces?

Conditions

• $V$ and $W$ are inner product spaces.
• $F$ is a vector space isomorphism.
• $F$ preserves inner products for all $a, b \in V$.

Reasoning, step by step

1. Identify the domain $V$ and codomain $W$ as inner product spaces.
2. Define the map $F: V \to W$.
3. Require $F$ to be a vector space isomorphism (preserving addition and scalar multiplication).
4. Require $F$ to preserve inner products: $\langle a, b \rangle_V = \langle F(a), F(b) \rangle_W$.

Example

The video states: 'Let V, W be inner product spaces. A map F: $V \to W$ is called a linear isometry if it is a vector space isomorphism that preserves inner products.'

Common misconceptions

• Thinking that any linear map is an isometry; it must also preserve inner products.
• Confusing linear isometry with affine isometry; linear isometry fixes the origin.

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