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Linear Transformations

  1. Oct 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Is the function which rotates the xy-plane by 20 degrees is a linear transformation?

    From R2 -> R2

    2. Relevant equations

    x` = xcos[tex]\theta[/tex] + ysin[tex]\theta[/tex]
    y` = -xsin[tex]\theta[/tex] + ycos[tex]\theta[/tex]

    Where [tex]\theta[/tex] = 20 degrees (or [tex]\pi[/tex]/9 )


    3. The attempt at a solution
    Apparently the solution to this is true:

    Ok so the two properties must hold

    T(u + v) = T(u) + T(v)
    and T(cu) = cT(u)
    ------------------
    Let u = (u1, u2) and v =(v1, v2)


    T(u) + T(v) = u1cos[tex]\theta[/tex] + u2sin[tex]\theta[/tex], - v1sin[tex]\theta[/tex] + v2cos[tex]\theta[/tex]

    However this isn't equal to T(u+v) = T((u1 + v1), (u2 + v2)) which when expanded will give me more cos[tex]\theta[/tex] terms than in T(u) + T(v).

    I figure im just not computing this properly, any help would be great, thanks.
     
  2. jcsd
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