# Difference between orthogonal transformation and linear transformation

What is the difference between orthogonal transformation and linear transformation?

## Answers and Replies

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

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.

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

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.

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.

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:
Stephen Tashi
Orthogonal transformation $\subset$ Linear transformation