Orthogonal linear maps are generalizations of orthogonal matrices. By definition, a linear map \(f\) over an inner product real vector space \(V\) is an orthogonal linear map (function, or transformation) if it preserves the inner product, i.e. for all vectors \(u\) and \(v\) in \(V\), we have $$ f(u) \cdot f(v) = u \cdot v.$$

Orthogonal linear maps are 1-to-1 (injective) and the proof is as follows:

Let \(f\) be an orthogonal linear map and \(f(u) = 0\). From these assumptions, we see that $$ u \cdot u = f(u) \cdot f(u) = 0.$$ This implies that \(u = 0\). So, the kernel of \(f\) is the singleton \(\{0\}\), and so, by a property of 1-to-1 linear maps discussed in the post on 1-to-1 and onto linear maps, \(f\) is 1-to-1.

Now, we proceed to investigate the basic properties of orthogonal linear maps:

### Basic properties of orthogonal linear maps

In the following, we investigate the basic properties of orthogonal linear maps:

**Result**. Orthogonal linear maps preserve the norm induced by inner products. In other words, if \(V\) is an inner product real vector and \(f\) is a linear map from \(V\) to \(V\), then for all \(v\in V\), we have $$\Vert f(v) \Vert = \Vert v \Vert.$$

The proof is as follows: $$ \Vert v \Vert^2 = v \cdot v = f(v) \cdot f(v) = \Vert f(v) \Vert^2.$$ Since \(\Vert f(v) \Vert\) and \(\Vert v \Vert\) are non-negative numbers, we will obtain the desired result.

**Result**. Orthogonal linear maps preserve distances, i.e. orthogonal linear maps are isometries.

In view of the latter result the proof goes as follows: $$\Vert f(u) – f(v) \Vert = \Vert f(u-v) \Vert = \Vert u – v \Vert.$$

The proof of the following is left to the reader:

**Exercise**. Prove that orthogonal linear maps preserve the angles between vectors.

The reader is familiar with the concept of groups in abstract algebra should be able to prove the following:

**Exercise**. The set of all orthogonal linear maps over an inner product finite-dimensional vector space \(V\) equipped with the composition of functions is a group.

#### Examples of orthogonal linear maps

Orthogonal matrices are examples of orthogonal linear maps. In fact, linear maps over finite-dimensional inner product real vector spaces are orthogonal if and only if their matrices are orthogonal.