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Matrix of a linear transformation

  1. Mar 23, 2009 #1
    Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.

    Orthogonal Projection T onto the line in R^3 spanned by
    (1 1 1)


    I'm assuming (though I tend to be wrong) that I need to find a vector that is parallel to the line and 2 that are perpendicular to it and linearly independent.

    So would the B matrix look something like this

    1 -2 1

    1 1 0

    1 1 -1

    ?

    Or am I once again way off track :)
     
  2. jcsd
  3. Mar 24, 2009 #2

    lanedance

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    Hi Succubus
    misread question... run with mazes comments
     
    Last edited: Mar 24, 2009
  4. Mar 24, 2009 #3
    Looks good. A few follow up questions worth thinking about would be,
    1) If you write a vector in the basis B, v = v1 (1 1 1)T + v2 (-2 1 1)T + v3 (1 0 -1)T, then what is Tv in this basis?
    2) What is the matrix T in the basis B?
    3) What would go wrong if the second vector in B was not perpendicular to the first one?

    *extra challenge*4) What is T in the standard basis (1,0,0)T, (0,1,0)T, (0,0,1)T? Can you write the projection matrix in terms of an outer product (ie: T = q qT for some vector q)?
     
    Last edited: Mar 24, 2009
  5. Mar 24, 2009 #4
    Thanks, I appreciate your help. Now, what is it was a reflection instead of a projection? Let's say a reflection about the plane (1 1 1) Would I do the same procedure?
     
  6. Mar 24, 2009 #5

    lanedance

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    hi succubus - do you mean the reflection in the plane through the origin, with normal vector (1,1,1) ?

    if so, you can look at your previous vectors (2 perpindicular and 1 parallel to (1,1,1)).

    Say v1= (1,1,1), this will be mapped to (-1,-1,-1). Whilst the perpindicular vectors (v2,v3) will be mapped to themselves,as they are in the plane of reflection

    so you know
    [tex] A.v_1 = -v_1 [/tex]
    [tex] A.v_2 = v_2 [/tex]
    [tex] A.v_3 = v_3 [/tex]

    can you write A in the basis of v1,v2,v3? then you can you transform to the original basis if needed...
     
  7. Mar 24, 2009 #6
    Thanks! But the problem is

    Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.

    Reflection T about the line in R^3 spanned by
    (1 1 1)


    I accidentally put plane! I'm sorry.

    So I would find 2 vectors perpendicular to the line and 1 parallel? I think I see it, but I don't understand exactly why. :/

    So it would be

    1 -1 1
    1 0 -1
    1 1 0

    ?
     
  8. Mar 24, 2009 #7

    lanedance

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    hmm.. ok, so i assume it will reflect any vector u, in a plane parallel to (1,1,1)
    and if we write
    [tex] u = v+ \lambda(1,1,1) [/tex] for some unqiue [tex] v, \lambda [/tex] for every u
    v will be normal to the plane of reflection...(someone correct me if i am wrong)

    as before, say you have vectors
    v1 - parallel (1,1,1)
    v2 - perp (1,1,1)
    v3 - perp (1,1,1) (v3 not equal v2)

    in this case
    [tex] A.v_1 = v_1 [/tex]
    [tex] A.v_2 = -v_2 [/tex]
    [tex] A.v_3 = -v_3 [/tex]

    then comments as before
     
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