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Math Help - Scale Factor

  1. #1
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    Scale Factor

    hello,

    can someone explain me what a volume scale factor is and if I have a liner Transformation
    T(v1) = av1 + bv2
    T(v2) = cv1 + dv2

    how can I find the volume scale factor of it?

    Thank you
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  2. #2
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    You mean area factor? (Volume factors are for 3D...)

    Think of it this way: if you have an enlargement matrix, say scale factor 2, then your matrix is \begin{bmatrix}2 & 0\\0 & 2\end{bmatrix}.

    So, if we define a unit square (1x1 length) using four points, say A = \begin{bmatrix}0\\0\end{bmatrix}, B = \begin{bmatrix}0\\1\end{bmatrix}, C = \begin{bmatrix}1\\1\end{bmatrix} and D = \begin{bmatrix}1\\0\end{bmatrix}.

    Then if we put all those points into the matrix, the transformed points are A' = \begin{bmatrix}0\\0\end{bmatrix}, B' = \begin{bmatrix}0\\2\end{bmatrix}, C' = \begin{bmatrix}2\\2\end{bmatrix} and D' = \begin{bmatrix}2\\0\end{bmatrix}.

    These points define a matrix with an area of 4 square units, so the original unit square got enlarged by an area factor of 4. That is to say, if you used points to define a polygon with area n square units, then the transformed shape will have an area of 4n square units.

    For any generic transformation matrix, the area factor is equal to the determinant.

    i.e: For any matrix M = \begin{bmatrix}a & b\\c & d\end{bmatrix}, the area factor is det(M) = ad-bc (for a 2x2 matrix).
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  3. #3
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    Thanks so much for the reply, i think I understood what you say , but no I meant volume scale factor or metric scale factor, that's how it was given to me.

    Regards
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  4. #4
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    Oh, oops- my bad, I thought you were dealing with 2x2 matrices only since your original post had a system of two vectors.

    In general, the volume factor is just the determinant of the matrix. (We say "volume factor" for anything above 2D.)
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