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www.mtsolitary.com
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stephenmalina.com
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| | | | | Selected Exercises # 5.A # 12. Define $ T \in \mathcal L(\mathcal P_4(\mathbf{R})) $ by $$ (Tp)(x) = xp'(x) $$ for all $ x \in \mathbf{R} $. Find all eigenvalues and eigenvectors of $ T $. Observe that, if $ p = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 $, then $$ x p'(x) = a_1 x + 2 a_2 x^2 + 3 a_3 x^3 + 4 a_4 x^4. | |
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stephenmalina.com
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| | | | | Selected Exercises # 4. Suppose $m$ and $n$ are positive integers with $ m \leq n $, and suppose $ \lambda_1, \dots, \lambda_m \in F $. Prove that there exists a polynomial $ p \in \mathcal P(\mathbf{F}) $ with $ \deg p = n $ such that $ 0 = p(\lambda_1) = \cdots = p(\lambda_m) $ and such that $ p $ has no other zeros. First, we can show that the polynomial $p'(z) = (z - \lambda_1) \cdots (z-\lambda_m) $ with $ \deg p' = m $ has $ 0 = p'(\lambda_1) = \cdots = p'(\lambda_m) $ and no other zeros. | |
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sansmap.wordpress.com
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| | | hand written text by shel silverstein, poem published in his book 'Falling Up'; 1996 | ||
