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Vector equation vs Matrix equation

  1. Feb 5, 2012 #1
    I am a little bit confused about these 2.
    A vector equation goes like this: x1v1+x2v2+...+xnvn=b
    and Matrix equation goes like this:
    Ax= [v1 v2 .. vn][x1
    x2
    .
    .
    .
    xn]

    v is a vector in the vector equation, but in the matrix equation x becomes the vector and v is just the matrix.

    My question is, why does it get switched like this?
     
  2. jcsd
  3. Feb 5, 2012 #2

    Mark44

    Staff: Mentor

    There are different kinds of multiplication going on. In your first equation, each term is a scalar times a vector (I'm assuming that the xi's are scalars).

    Your second equation is the product of what appears to be an n X n matrix and a vector with n components. If you switched the order in the 2nd equation to xA, the multiplication would not be defined.
     
  4. Feb 5, 2012 #3

    Deveno

    User Avatar
    Science Advisor

    in your "vector equation" the v's are vectors, and the x's are field elements (scalars).

    one way of thinking of a matrix is "a vector of vectors" (it's a 2-array, not a 1-array). so in your "matrix equation" each v is a column (which is why matrix entries need TWO subscripts, one for the row, and one for the column). so what you really have is:

    [tex]\begin{bmatrix}v_{11}&v_{12}&\dots&v_{1n}\\v_{21}&v_{22}&\dots&v_{2n}\\ \vdots&\vdots&\ddots&\vdots\\v_{m1}&v_{m2}&\dots&v_{mn}\end{bmatrix} \begin{bmatrix}x_1\\x_2\\ \vdots\\x_n\end{bmatrix} = \begin{bmatrix}v_{11}x_1 + v_{12}x_2 + \dots + v_{1n}x_n\\v_{21}x_1 + v_{22}x_2 + \dots + v_{2n}x_n\\ \vdots\\v_{m1}x_1 + v_{m2}x_2 + \dots + v_{mn}x_n\end{bmatrix} = \begin{bmatrix}b_1\\b_2\\ \vdots\\b_n\end{bmatrix}[/tex]

    writing Ax = b is just an abbreviation for that god-awful mess above.
     
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