Difference between orthogonal transformation and linear transformation

1. Aug 4, 2013

EnglsihLearner

What is the difference between orthogonal transformation and linear transformation?

2. Aug 4, 2013

micromass

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

3. Aug 5, 2013

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.

4. Aug 5, 2013

micromass

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

5. Aug 5, 2013

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.

6. Aug 5, 2013

micromass

Staff Emeritus
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.

7. Aug 5, 2013

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.

Last edited: Aug 5, 2013
8. Oct 29, 2014

9. Nov 2, 2014

Stephen Tashi

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.

10. Nov 3, 2014

Orthogonal transformation $\subset$ Linear transformation