12 in the case where a is a symmetric n n matrix and

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Unformatted text preview: 1)x2 + 2x2 = 1. 2 2.2.2. Suppose that A= 2 −2 42 −2 5 . 2 2 0 1 -2 -2 -2 0 -1 -1 0 1 2 -2 Figure 2.4: Hyperboloid of two sheets. a. Find an orthogonal matrix B such that B T AB is diagonal. b. Sketch the conic section 2x2 − 4x1 x2 + 5x2 = 1. 1 2 c. Sketch the conic section 2x2 − 4x1 x2 + 5x2 − 4x1 + 4x2 = −1. 1 2 2.2.3. Suppose that A= 4 2 2 1 . a. Find an orthogonal matrix B such that B T AB is diagonal. √ √ b. Sketch the conic section 4x2 + 4x1 x2 + x2 − 5x1 + 2 5x2 = 0. 1 2 2.2.4. Determine which of the following conic sections are ellipses, which are hyperbolas, etc.: a. x2 + 4x1 x2 + 3x2 = 1. 1 2 b. x2 + 6x1 x2 + 10x2 = 1. 1 2 c. −3x2 + 6x1 x2 − 4x2 = 1. 1 2 d. −x2 + 4x1 x2 − 3x2 = 1. 1 2 2.2.5. Find the semi-major and semi-minor axes of the ellipse 5x2 + 6x1 x2 + 5x2 = 4. 1 2 43 2.2.6. Suppose that 0 A= 2 0 2 3 0 0 0 . 9 a. Find an orthogonal matrix B such that B T AB is diagonal. b. Sketch the quadric surface 4x1 x2 + 3x2 + 9x2 = 1. 2 3 2.2.7. Determine w...
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This document was uploaded on 01/12/2014.

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