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    Does it matter how you change vectors into a matrix?

    Let's say you were given
    (2,1,1), (3,5,9),(5,4,3)

    and you had to figure out the inverse or L.I
    would it matter if you made the matrix into:
    (2,1,1)
    (3,5,9)
    (5,4,3)

    rather than:
    (2,3,5)
    (1,5,4)
    (1,9,3)

    So for one I turned the vectors into the columns of the matrix and the other into rows. In my example book it does it in different ways and keeps interchanging
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    (Original post by 0range)
    Does it matter how you change vectors into a matrix?

    Let's say you were given
    (2,1,1), (3,5,9),(5,4,3)

    and you had to figure out the inverse or L.I
    would it matter if you made the matrix into:
    (2,1,1)
    (3,5,9)
    (5,4,3)

    rather than:
    (2,3,5)
    (1,5,4)
    (1,9,3)

    So for one I turned the vectors into the columns of the matrix and the other into rows. In my example book it does it in different ways and keeps interchanging
    Yes it does matter, I would say that the second method is more common. I can't think when you would need to do this (I am a bit rusty though).
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    (Original post by lizard54142)
    Yes it does matter, I would say that the second method is more common. I can't think when you would need to do this (I am a bit rusty though).
    I keep getting conflicting answers I use different resources when I study and it seems that they do it differently.

    You'd do this, say, to find the kernal of a set of vectors or the columnspace or whatever
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    I may be wrong but think about it this way,

    You have co-ordinates (1,2,3), (4,5,6) and (7,8,9). If you put them into the matrix
    (1,2,3)
    (4,5,6)
    (7,8,9)

    You then multiply the matrix by (X)
    (Y)
    (Z)
    You will find that that you will end up with 1X+2Y+3Z, I know this a line equation not a vector but it shows how the X y and z dimensions are mapped to each column. You could do it vice versa, so you have

    (1,4,7)
    (2,5,8)
    (3,6,9)

    Then multiply (X,Y,Z) by this matrix and you will end up with the same 1X+2Y+3Z

    So it can be either way as long as you are consistent throughout.

    I didn't explicitly do vectors to matrix but I think this should help you understand better.

    Good luck

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    You can do everything perfectly sensibly both ways. To explain things, I'm going to have to go over the fundamentals of what matrices are properly, since you probably didn't do it properly at school:

    A matrix is a way of representing a linear map. A linear map is a function f between two spaces (in our case, from \mathbb{R}^n to itself) such that for all x, y in the space, and for all a \in \mathbb{R} (you can do this with complex numbers instead, but I'll stick to the reals since everything that I'll talk about is exactly the same in both cases), f(x+y) = f(x) + f(y), and f(ax) = af(x).

    The three vectors that you are constructing your matrix out of are the images of your basis vectors (we'll choose our basis to be (1,0,0), (0,1,0), (0,0,1), since we can just relabel things to make any other sensible (meaning orthonomal, for the pedants out there) basis the same as this anyway, and it makes everything neater.

    There choice between the two was that you have constructed your matrix basically comes down to whether you prefer left-multiplication (with row vectors), or right-multiplication (with column vectors).


    To explain that better: your two matrices represent the same linear map, but written differently. With A being our matrix, and f the linear map that it represents, putting the vectors in as columns is f(x,y,z) = A\left(\begin{array}{c} x \\ y \\ z \end{array}\right), whereas putting them in as rows is f(x,y,z) = (x,y,z)A. Everything else will be the same: you'll get the same determinant (since it's the same linear map), the same kernel (since it's the same linear map), and the inverse (should it exist) will be the matrix of the inverse map written in each of these different ways (in particular, the inverse obtained in one way will be the transpose of the one obtained the other way).

    Most people prefer putting them in as columns because f(x,y,z) = A\left(\begin{array}{c} x \\ y \\ z \end{array}\right) looks more natural than f(x,y,z) = (x,y,z)A
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    (Original post by 0range)
    would it matter if you made the matrix into:
    (2,1,1)
    (3,5,9)
    (5,4,3)

    rather than:
    (2,3,5)
    (1,5,4)
    (1,9,3)
    Note that your matrices are transposes of each other. It is a result of linear algebra that:

    (A^{-1})^T = (A^T)^{-1}

    if A is invertible. So it doesn't matter whether you write the vectors in column or row form when you create the matrix since if you decide, after finding the inverse, that you really want the inverse of the transposed matrix, you can just take the transpose of the inverse to get it.
 
 
 
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