Orthogonal Transformation

Topics: Algebra - Vector Space - Linear Transformation

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(definition)

Let $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear transformation.

We say that $T$ is orthogonal if:

$$⟨x,y⟩=⟨T(x),T(y)⟩\forall x,y\in {\mathbb{R}}^n$$

(theorem)

If $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is linear, then $T$ is orthogonal if and only if (norm):

$$\Vert T(x)\Vert =\Vert x\Vert \forall x\in {\mathbb{R}}^n$$

Example

Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be defined as $T(x,y,z):=(x,yz)$ (the identity transformation).

Is $T$ orthogonal?

For $(x,y,z)\in {\mathbb{R}}^3$, we have that:

$$⟨(x,y,z),(x,y,z)⟩=x^2+y^2+z^2$$

$$⟨T(x,y,z),T(x,y,z)⟩=x^2+y^2+z^2$$

Voilà, that tells us that $T$ is orthogonal>