Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² + 2bxy+ay² = c with b 0 and c> 0. a (öö). b Let V(x, y) = ax² +2bry + ay² and denote A = Note that: v (2, 1) = (2 v) A (*) = ((*), 4 (*)) А where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis ü, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector (+) ER2 as a linear combination of u and . (c) Use result from (b) to transform (z,y) = ((*). 4 ()) A some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into V(u, v) = au² + Bv² for

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