Stitz-Zeager_College_Algebra_e-book

Focusing on the rst system we have 1 1 2 312 3 1 1 5

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Unformatted text preview: omials involving matrices. The reader is encouraged to step back and compare our expansion of the matrix product (M − 2I4 ) (M + 3I4 ) in part 3 with the product (x − 2)(x + 3) from real number algebra. The exercises explore this kind of parallel further. As we mentioned earlier, a point P (x, y ) in the xy -plane can be represented as a 2 × 1 position matrix. We now show that matrix multiplication can be used to rotate these points, and hence graphs of equations. √ Example 8.3.3. Let R = 2 2 √ 2 2 √ − 2 2 √ 2 2 . 1. Plot P (2, −2), Q(4, 0), S (0, 3), and T (−3, −3) in the plane as well as the points RP , RQ, RS , and RT . Plot the lines y = x and y = −x as guides. What does R appear to be doing to these points? 2 2. If a point P is on the hyperbola x2 − y 2 = 4, show that the point RP is on the curve y = x . 486 Systems of Equations and Matrices Solution. For P (2, −2), the position matrix is P = 2 , and −2 √ RP = = 2 2 √ 2 2 √ √ − 2 2 √ 2 2 2 −2 22 0 √ √√ We have that R √ √(2, −2) to (2 2, 0). Similarly, we find (4, 0) is moved to (2 2, 2 2), (0, 3) takes ...
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