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P. Moller - Maersk. com 56 Linear Algebra Examples c-3 Example dx1 dt dx2 dt dx3 dt 2. 3 Solve the system of differential equations = 4x1 (t) + 2x2 (t) + 2x3 (t), = 2x1 (t) + 4x2 (t) + 2x3 (t), = 2x1 (t) + 2x2 (t) + 4x3 (t). The corresponding ⎛ 4 2 A=⎝ 2 4 2 2 matrix is ⎞ 2 2 ⎠. 1 If we add the three equations, then we immediately get d {x1 (x) + x2 (t) − x3 (t)} = 8{x1 (t) + x2 (t) + x3 (t)}, dt and analogously by a subtraction, d {x1 (t) − x2 (t)} = 2x1 (t) − 2x2 (t) = 2{x1 (t) − x2 (t)}, dt and d {x2 (t) − x3 (t)} = 2x2 (t) − 2x3 (t) = 2{x2 (t) − x3 (t)}.

Prove that if x = 2. In particular, let M= 1 2 2 1 , and let f : C4 → C4 be the linear map, which with respect to the usual basis is given by the matrix A, corresponding to M. Find the eigenvalues and the eigenvectors of f . 3. Find a basis (b1 , b2 , b3 , b4 ) of C4 , such that the matrix of f with respect to this basis is a diagonal matrix, and find this diagonal matrix. ⎛ ⎞ x ⎜ y ⎟ ⎟ 1. When we put z = ⎜ ⎝ μx ⎠, then μy ⎛ ⎞⎛ 0 0 1 0 x ⎜ 0 0 0 1 ⎟⎜ y ⎟⎜ Az = ⎜ ⎝ a b 0 0 ⎠ ⎝ μx c d 0 0 μy ⎞ ⎛ ⎞ ⎛ μx ⎟ ⎜ μy ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎠ ⎝ λx ⎠ = μ ⎝ λy ⎞ x y ⎟ ⎟ = μz, μx ⎠ μy and the claim is proved.

A3 a 1 a3 · a 2 a3 · a 3 We note that this matrix is always symmetric, because the inner product is symmetric. com 62 Linear Algebra Examples c-3 3. 5 Consider in the vector space P2 (R) of all real polynomials of at most degree 2 with the scalar product 1 where P (x), Q(x) ∈ P2 (R). P (x)Q(x) dx, P, Q = −1 Find the angles between the vectors below in P2 (R): P1 (x) = 1, P3 (x) = 1 − x. P2 (x) = x, It follows from P1 2 P2 2 P3 2 1 −1 12 dx = 2, x3 1 = −1 x2 dx = 3 = = 1 (1 −1 P1 = 1 −1 2 = , 3 − x)2 dx = 2 0 1 x2 2 P2 u2 du = u3 3 2 = 0 8 , 3 P3 √ 2, 2 , = 3 2 , =2 3 and P 1 , P2 = P 1 , P3 = −1 1, 1 − x = P1 1 P 2 , P3 1 · dx = = −1 1 −1 2 = 0 = P1 · P2 cos(∠(P1 , P2 )), − P1 , P1 2 = 2 = P1 · P3 cos(∠(P1 , P3 )), x · (1 − x) dx = x2 x3 − 2 3 1 −1 =− 2 = P2 · P3 cos(∠(P2 , P3 )), 3 DIVERSE - INNOVATIVE - INTERNATIONAL Please click the advert Are you considering a European business degree?

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Absolutely summing operators from the disc algebra


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