Complexity Explorer Santa Fe Institute

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Vector and Matrix Algebra

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3.2 Geometric Transformations » Quiz #19 Solution

Question 1:

Recall that the matrix encoding the linear transformation of a reflection about the x-axis is given by: A_x=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. Because a succession of linear transformations can be encapsulated through a product of matrices (ordered right-to-left), the sequence of a counter-clockwise rotation by 30 degrees, followed by a reflection about the x-axis, is written as a single matrix equal to the product of the respective transformation matrices. 

Accordingly, A=A_xA_{30}=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} \frac{\sqrt3}{2}} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt3}{2} \end{bmatrix} =\begin{bmatrix} \frac{\sqrt3}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt3}{2} \end{bmatrix}.