Student 999
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How do I do question 2Name:  76B3D21B-14F7-43DE-A5E2-A3AF9919A800.png
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RDKGames
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(Original post by Student 999)
How do I do question 2
Note that the definition is equivalent to

\begin{pmatrix} a_{n+1} \\ b_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1\end{pmatrix} \begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix}

And the one you need is

\begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix} = \begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1\end{pmatrix}^{-1} \begin{pmatrix} a_{n+1} \\ b_{n+1} \end{pmatrix}
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Student 999
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(Original post by RDKGames)
Note that the definition is equivalent to

\begin{pmatrix} a_{n+1} \\ b_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1\end{pmatrix} \begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix}

And the one you need is

\begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix} = \begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1\end{pmatrix}^{-1} \begin{pmatrix} a_{n+1} \\ b_{n+1} \end{pmatrix}
I thought it was a 2 by 1 matrix since they included the addition sign
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DFranklin
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(Original post by Student 999)
I thought it was a 2 by 1 matrix since they included the addition sign
What they've *written* is a 2 x 1 matrix, but the *operation* it's describing can be represented by a 2 x 2 matrix.

For a general 2 x 2 matrix, the definition of multiplication is

\begin{pmatrix}a & b \\ c & d \end{pmatrix} \begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}

The *result* of the matrix multiplication is \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}, which is a 2 x 1 matrix, but the actual *operation* (i.e. the thing you do to \begin{pmatrix}x \\ y \end{pmatrix} to get  \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}) is defined by the 2 x 2 matrix with entries a,b,c,d.
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mqb2766
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(Original post by Student 999)
How do I do question 2Name:  76B3D21B-14F7-43DE-A5E2-A3AF9919A800.png
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A slightly different way of approaching b) is to think about how a and b are transformed by each iteration.
[a,0] -> [a, a*sqrt(3)]
[0,b] -> [-b*sqrt(3), b]
Drawing the transformations as triangles, it should be fairly obvious how a general vector is expanded and rotated. The inverse iterations are then rotation in the opposite direction and compression.

It should explain why they ask you to do 6 (inverse) transformations and why the 7th term is powers of 2.
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