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Apparent fallacy in linear operator theory

  1. Jan 8, 2009 #1
    Butkov's book present the theory of linear operators this way:

    Suppose a linear operator [tex]\alpha[/tex] transforms a basis vector
    [tex]\hat{\ e_i}[/tex] into some vector [tex]\hat{\ a_i}[/tex].That is we have

    [tex]\alpha\hat{\ e_i}=\hat{\ a_i}[/tex]....................(A)

    Now the vectors [tex]\hat{\ a_i}[/tex] can be represented by its co-ordinates w.r.t. basis [tex]\{\hat{\ e_1},\hat{\ e_2}, ...,\hat{\ e_N}}[/tex].

    [tex]\hat{\ a_i} = \sum\ a_j_i\hat{\ e_j}[/tex] where i,j=1,2,3...N and summation over j is implied.................(B)

    Notice that the in last equation,we have put a row vector(a)=a row vector (e) times a matrix A

    Now with the help of the transforming matrix [tex]\ a_j_i[/tex],we can find the co-ordiantes of [tex]\ y=\alpha\ x[/tex] from the co-ordiantes of [tex]\ x[/tex]

    [tex]\ y=\alpha\ x=\alpha\sum [\ x_i\hat{ e_i}]=\sum [\ x_i\hat{ a_i}][/tex]................(C)

    Employing the definition of [tex]\ a_i[/tex] as in (B), we obtain

    [tex]\ y= \sum\ x_i\sum\ a_j_i\hat{\ e_j} = \sum[\sum\ a_j_i\ x_i]\hat{\ e_j}[/tex] in the last term the outer summation is on j.....................(D)

    From this we could identify that [tex]\ y=\sum\ y_j\hat{\ e_j}[/tex]....(E)

    where [tex]\ y_j=\sum[\ a_j_i\ x_i}[/tex]...............(F)

    Last equation shows y and x are column vectors.If they were row vectors, the indices of [tex]\ a[/tex] would have interchanged among themselves.

    But our very first assumption was [tex]\hat{ a_i}[/tex] is a row vector.And y is a linear combination of [tex]\hat{ a_i}[/tex].Thus, y should be a row vector!!!

    Can anyone please help me to see where is the fallacy?

  2. jcsd
  3. Jan 11, 2009 #2
    there is a subtle, but important difference between a vector and the matrix of components that represent the vector. Incidentally, the same thing can be said about a linear operator and the matrix of components representing the operator. The vector you express in (B) is a linear combination of the basis vectors you list in the line above (B). The matrix representing this vector is usually written as a Nx1 array, or column vector. Does this help at all?
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