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emilygorcenski.com | ||
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stephenmalina.com
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| | | | | Intro # I've recently been making my way through Axler's Linear Algebra Done Right and, as a way to motivate myself to continue, have decided to blog my notes and solutions for exercises as I go. Insights # Section 2.A # You can convert any linearly dependent list to a linearly independent list with the same span. # By the linear dependence lemma, if you have a list that's linearly dependenty, then you can remove one item without changing the list's span. | |
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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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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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pfasproject.com
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| | | Suspected source: Dupont's Washington Works plant DuPont began using PFOA to manufacture Teflon at its Washington Works plant in 1951. The company knew that PFOA is toxic in 1961. In 1981, DuPont foundevidence of birth defects in babies born to female employees who worked in its West Virginia plant, and decides topull female employees from... | ||
