Origins of Matrices

We can use simple trigonometry to relate the coordinates of Q to those of P by expressing the Cartesian coordinates in terms of polar coordinates. Thus, the (x,y) coordinates of point P become  

If we now use the addition theorems for cosine and sine (see Volume 1, Section 2.3.3), we obtain the expansions  

Equation (4.11) describes the transformation of coordinates under an anticlockwise rotation by an angle, 0. This coordinate transformation is completely characterized by a square matrix. A, with the elements cos 0 and + sin 0, and the column matrices, r and r, involving the initial and final coordinates, respectively  

We can now use matrix notation to replace the two equations (4.11) by the single matrix equation (4.13)  

We can confirm that equation (4.13) correctly represents the coordinate transformation by evaluating the product matrix Ar on the right side  

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