# 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