Difference between orthogonal transformation and linear transformation

What do you understand by a linear transformation and by an orthogonal transformation?

 

Do you know what a vector space is? Did you ever study linear algebra?

 

You should probably study linear algebra if you really want to grasp this.

I'll explain it for Euclidean spaces. A function ##T:\mathbb{R}^n\rightarrow \mathbb{R}^m## is called linear if the following two properties are satisfied

1) ##T(\mathbf{x} + \mathbf{y}) = T(\mathbf{x}) + T(\mathbf{y})## for ##\mathbf{x},\mathbf{y}\in \mathbb{R}^n##.
2) ##T(\lambda\mathbf{x}) = \lambda T(\mathbf{x})## for ##\mathbf{x}\in \mathbb{R}^n## and ##\lambda\in \mathbb{R}##.

Now, an orthogonal transformation is a linear transformation if it preserves the inner product. On ##\mathbb{R}^n## you have the inner product

[tex]\mathbf{x}\cdot \mathbf{y} = x_1 y_1 + ... + x_n y_n[/tex]

Thus an orthogonal transformation satisfies ##T(\mathbf{x}) \cdot T(\mathbf{y}) = \mathbf{x} \cdot \mathbf{y}## for each ##\mathbf{x},\mathbf{y}\in \mathbb{R}^n##. Note that by definition an orthogonal transformation is linear.

 

Here's a post that explains PCA in lay terms without going into the rigorous math. http://georgemdallas.wordpress.com/...nvectors-eigenvalues-and-dimension-reduction/

 

In 2D, an intuitive way to look at it is that linear transformations preserve parallelograms. Othogonal transformations preserve rectangles.

For example in 2D plane, one property of a linear transformation is that it preserves the origin of the plane and preserves those parallelograms that have one vertex at the origin. For example, it would be OK for a linear transformation to send the rectangle (0,0)(2,0),(2,1)(0,1) to the parallelogram with vertices (0,0),(2,0),(3,2)(1,2). An orthogonal transformation preserves rectangles. So it will not transform a rectangle in to a non-rectangular parallelogram. For example, a rotation of the plane by 30 degrees about the origin preserves such rectangles. The rotation also preserves parallelograms, so it is both a linear tranformation and an orthogonal transformation.

 

Orthogonal transformation [itex]\subset[/itex] Linear transformation