Difference between orthogonal transformation and linear transformation

[EnglsihLearner]

What is the difference between orthogonal transformation and linear transformation?

 

[micromass]

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

 

[EnglsihLearner]

When I start to learner PCA. I find the term "orthogonal transformation" unfamiliar. I google to to find the solution and I get anther unfamiliar term called "linear transformation". So I am unfamiliar with both the terms. I think if Can know the difference between them then it would be very helpful to understand the both term.

 

[micromass]

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

 

[EnglsihLearner]

Perhaps I studied Matrix if it is linear algebra. And I think I understand what is vector space.

Vector space:
http://en.wikipedia.org/wiki/Vector_space

Is it sufficient?

I got the definition of both terms by wikipedia. But I don't understand clearly.

 

[micromass]

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

$$\mathbf{x}\cdot \mathbf{y} = x_1 y_1 + ... + x_n y_n$$

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.

 

[EnglsihLearner]

Thanks.
I will back again after reading linear algebra. I am working on a topic called ECG(Electrocardiogram). I must understand PCA(Principal Component Analysis) to grasp ECG.


I hope with your help I will be able to understand PCA.

Thanks again.

 

[Alderaan]

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/

 

[Stephen Tashi]

What is the difference between orthogonal transformation and linear transformation?

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.

 

[JonnyMaddox]

Orthogonal transformation $\subset$ Linear transformation