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Difference between orthogonal transformation and linear transformation

  1. Aug 4, 2013 #1
    What is the difference between orthogonal transformation and linear transformation?
  2. jcsd
  3. Aug 4, 2013 #2
    What do you understand by a linear transformation and by an orthogonal transformation?
  4. Aug 5, 2013 #3
    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.
  5. Aug 5, 2013 #4
    Do you know what a vector space is? Did you ever study linear algebra?
  6. Aug 5, 2013 #5
    Perhaps I studied Matrix if it is linear algebra. And I think I understand what is vector space.

    Vector space:

    Is it sufficient?

    I got the definition of both terms by wikipedia. But I don't understand clearly.
  7. Aug 5, 2013 #6
    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.
  8. Aug 5, 2013 #7
    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
  9. Oct 29, 2014 #8
  10. Nov 2, 2014 #9

    Stephen Tashi

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    Science Advisor

    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.
  11. Nov 3, 2014 #10
    Orthogonal transformation [itex]\subset[/itex] Linear transformation
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