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# Download PDF by Mejlbro L.: Linear algebra c-3 - The Eigenvalue Problem and Euclideam

By Mejlbro L.

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Does there exist another basis of R3 , such that the matrix of f with respect to this new basis is a diagonal matrix? 3. Let v1 = (1, −1, 1) and v2 = (1, −2, 2). Find a vector v3 , such that f (v3 ) = v2 + v3 , and prove that (v1 , v2 , v3 ) form a basis of R3 . 4. Find the matrix of f with respect to the basis (v1 , v2 , v3 ). 1. We ﬁrst compute det(A − λI) = = 4 5−λ −4 4 1 1−λ 1 1−λ 0 −1 2 −1 − λ −1 2 − (λ + 1) 5−λ −4 1 1−λ = 4{2 + 1 − λ} − (λ + 1){(λ − 1)(λ − 5) + 4} = −4(λ − 3) − (λ + 1){λ2 − 6λ + 9} = −4(λ − 3) − (λ + 1)(λ − 3)2 = −(λ − 3){(λ + 1)(λ − 3) + 4} = −(λ − 3){λ2 − 2λ + 1} = −(λ − 1)2 (λ − 3).

The complete solution ⎞ ⎛ ⎛ x1 (t) ⎝ x2 (t) ⎠ = ⎝ x3 (t) is 0 et 0 e2t −2e2t −2e2t ⎞⎛ ⎞ e3t c1 −e3t ⎠ ⎝ c2 ⎠ , −e3t c3 where c1 , c2 , c3 are arbitrary constants. com 55 Linear Algebra Examples c-3 2. Systems of differential equations 2. If we put t = 0, then ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎞⎛ x1 (0) −1 0 1 1 c1 ⎝ x2 (0) ⎠ = ⎝ 1 −2 −1 ⎠ ⎝ c2 ⎠ = ⎝ 1 ⎠ . c3 x3 (0) 0 0 −2 1 It follows from the reductions ⎞ ⎛ ⎛ −1 0 1 0 1 1 ⎝ 1 −2 −1 1 ⎠∼⎝ 1 0 0 2 0 0 −2 1 1 0 −1 ⎞ ⎛ −1 1 0 0 1 ⎠∼⎝ 0 1 0 0 0 0 1 1 ⎞ − 13 ⎠ − 23 1 2 and c3 = − , hence we get the solution 3 3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ 3t ⎞ ⎞ x1 (t) 0 e2t e 1 1 ⎝ x2 (t) ⎠ = ⎝ et ⎠ − ⎝ −2e2t ⎠ − ⎝ −e3t ⎠ .

Besides being a manager in the Manufacturing IT department, Kim performs triathlon at a professional level. ‘NNE Pharmaplan offers me freedom with responsibility as well as the opportunity to plan my own time. com NNE Pharmaplan is the world’s leading engineering and consultancy company focused exclusively on the pharma and biotech industries. NNE Pharmaplan is a company in the Novo Group. com 33 Linear Algebra Examples c-3 1. g. (1, −1, 1) = v1 . 2. Now, λ = 1 has the algebraic multiplicity 2 and the geometric multiplicity 1.