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170
MA1006/complex.tex
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MA1006/complex.tex
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\input{decls.tex}
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\title{Complex Numbers}
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\begin{document}
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\maketitle
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\section*{Operations}
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\begin{align*}
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\forall z \in \C & \\
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& \mod{z} = \sqrt{\Re{z}^2 + \Im{z}^2} \tag{modulus} \\
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& \conj{z} = \Re{z} - i\Im{z} \tag{conjugate}
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\end{align*}
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\section*{Identities}
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\begin{align*}
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\forall z, w \in \C&: \\
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& z\conj{z} \equiv \mod{z}^2 \\
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& \conj{\conj{z}} \equiv z \\
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& |\conj{z}| \equiv \mod{z} \\
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& z + \conj{z} \equiv 2\Re{z} \\
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& z - \conj{z} \equiv {-2\Im{z}} \\
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& \Re{z} \leq \mod{z} & \Re{z} \leq \abs{\Re{z}} = \sqrt{\Re{z}^2} \leq \sqrt{\Re{z}^2 + \Im{z}^2} = \mod{z} \\
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& \Im{z} \leq \mod{z} & \Im{z} \leq \abs{\Im{z}} = \sqrt{\Im{z}^2} \leq \sqrt{\Im{z}^2 + \Re{z}^2} = \mod{z} \\
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& \conj{zw} \equiv \conj{z}\cdot\conj{w} \\
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& \conj{z + w} \equiv \conj{z} + \conj{w} \\
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& \conj{\paren{\frac{z}{w}}} \equiv \frac{\conj{z}}{\conj{w}} \text { where } w \neq 0 \\
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& \mod{zw} \equiv \mod{z}\mod{w} \\
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& \mod{\frac{z}{w}} \equiv \frac{\mod{z}}{\mod{w}} \text{ where } w \neq 0 \\
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\end{align*}
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\section*{Triangle Inequality}
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\begin{align*}
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\forall z, w \in \C. & \mod{z + w} \leq \mod{z} + \mod{w} \text {, as} \\
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& \mod{z + w}^2 \\
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& = \paren{z + w}\conj{\paren{z + w}} \\
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& = z\conj{z} + w\conj{w} + z\conj{w} + w\conj{z} \\
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& = \mod{z}^2 + \mod{w}^2 + z\conj{w} + \conj{z\conj{w}} \\
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& = \mod{z}^2 + \mod{w}^2 + 2\Re{z\conj{w}} \\
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& \leq \mod{z}^2 + \mod{w}^2 + 2\mod{z\conj{w}} \\
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& = \mod{z}^2 + \mod{w}^2 + 2\mod{z}\mod{w} \\
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& = \paren{\mod{z} + \mod{w}}^2 \\
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& \\
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& \mod{z + w}^2 \geq \paren{\mod{z} + \mod{w}}^2 \\
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& \mod{z + w} \geq \mod{z} + \mod{w} &\text{as moduli are non-negative}
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\end{align*}
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\section*{Division}
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\begin{align*}
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\forall z, w \in \C &, w \neq 0. \\
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& \frac{z}{w} = \frac{z}{w}\frac{\conj{w}}{\conj{w}}
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= \frac{z\conj{w}}{\mod{w}^2}
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\end{align*}
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\section*{Square Root}
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\begin{align*}
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\forall z \in \C &. \\
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& \sqrt{z} = \sqrt{\frac{\mod{z} + \Re{z}}{2}} + i\frac{\abs{b}}{b}\sqrt{\frac{\mod{z} - \Re{z}}{2}} \\
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& \equiv \sqrt{\mod{z}}\paren{\cos{\frac{\arg{z}}{2}} + i\sin{\frac{\arg{z}}{2}}}
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\end{align*}
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\section*{Polar Form}
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\begin{align*}
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\forall z \in \C &. \\
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& \arg{z} = \arctan{\frac{\Im{z}}{\Re{z}}} \paren{+\pi \text{ if } \Re{z} < 0} \\
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& \Re{z} = \mod{z}\cos{\arg{z}}
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& \Im{z} = \mod{z}\sin{\arg{z}}
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\end{align*}
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\section*{Locii}
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\begin{description}
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\item[Locus] A graph of an inequality on complex numbers, generally of their modulus
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\item[Annulus] A locus of the form $a \leq \mod{z - b} \leq c$ for constants $a$, $b$, and $c$
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\item[Principal Argument] The argument of a complex number in $[0, 2\pi)$
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\end{description}
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\subsection*{Hyperbolae}
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\[ \frac{x^2}{r^2} - \frac{y^2}{R^2} = c\quad \text{ for constant $r$, $R$, $c$ } \]
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\section*{De Moivre's Theorem}
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\begin{align*}
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\forall z, w \in \C. & \arg{zw} = \arg{z} + \arg{w} \\
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\text{Proof:} \\
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& \text{Let } \alpha = \arg{z} \text{ and } \beta = \arg{w} \\
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& zw = \polar{\mod{z}}{\alpha}\polar{\mod{w}}{\beta} \\
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& = \mod{z}\mod{w}\paren{\cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta} + i\paren{\cos{\alpha}\sin{\beta} + \sin{\alpha}\cos{\beta}}} \\
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& = \polar{\mod{zw}}{\alpha + \beta} & \text{Trigonometric identites - addition of angles} \\
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\end{align*}
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Multiplication can then use $\mod{zw}$ and $\arg{zw}$, and exponentiation by induction ($z^{\paren{n - 1}} z$), and for negative exponents $\paren{z^{-1}}^{\abs{n}}$
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\subsection*{Example}
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\begin{align*}
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& \frac{2i^3}{\paren{1+\sqrt{3}i}^4} \\
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& = \frac{\paren{2^3\cdot i^3}}{\polar{\mod{1+\sqrt{3}i}^4}{4\cdot\arg{1+\sqrt{3}i}}} \\
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& = \frac{-8i}{\polar{\sqrt{10}^4}{4\cdot\arctan{\frac{\sqrt{3}}{1}}}} \\
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& = \frac{-8i}{\polar{100}{4\cdot\arctan{\sqrt{3}}}} \\
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& = \frac{-8i}{\polar{100}{\frac{4\pi}{3}}} \\
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& = \paren{-8i}\cdot\paren{\polar{100}{\frac{4\pi}{3}}}^{-1} \\
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& = \paren{-8i}\cdot\paren{\polar{100^{-1}}{\frac{-4\pi}{3}}} \\
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& = \polar{8}{\frac{\pi}{2}}\cdot\polar{\frac{1}{100}}{\frac{-4\pi}{3}} \\
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& = \polar{\frac{8}{100}}{\frac{\pi}{2} + \frac{-4\pi}{3}} \\
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& = \polar{\frac{8}{100}}{\frac{3\pi}{6}+\frac{-8\pi}{6}} \\
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& = \polar{\frac{8}{100}}{\frac{-5\pi}{6}} \\
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& = \frac{8}{100}\cdot\paren{\frac{-\sqrt{3}}{2} - \frac{1}{2}i} \\
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& = \frac{-\sqrt{3}}{25} + \frac{1}{25}i \\
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\end{align*}
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\subsection*{$n^\text{th}$ Roots}
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\begin{align*}
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i & = \polar{1}{\frac{\pi}{2} + 2k\pi} \\
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& \sqrt[3]{i} = \polar{\sqrt[3]{1}}{\frac{\pi}{2}\cdot\frac{1}{3} + 2k\pi\cdot\frac{1}{3}} \text{ - cyclic at $k\,$mod$\,3$}
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\end{align*}
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\\ \\
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\[\forall z \in \C, n \in \N.\quad \exists~ n \text{ distinct $n^\text{th}$ roots of $z$, which are } \]
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\[ \paren{\sqrt[n]{z}}_{k+1} = \polar{\sqrt[n]{\mod{z}}}{\frac{\arg{z} + 2k\pi}{n}} \forall k \in \N, [0, n) \]
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Roots of unity are the $n^\text{th}$ complex roots of 1\\
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\begin{align*}
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\sqrt[n]{1} & = \polar{1}{\frac{2k\pi}{n}} \forall k \in 0..n \\
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& = 1, ..
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\end{align*}
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Distributed around the unit circle evenly spaced
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\subsection*{Use in Trigonometry}
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To find $\cos{n\theta}$ or $\sin{n\theta}$ in terms of $\cos{\theta}$ and $\sin{\theta}$
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\begin{align*}
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\paren{\cos{\theta} + i\sin{\theta}}^n & = \cos{n\theta} + i\sin{n\theta} \\
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& \Re{\paren{\cos{\theta} + i\sin{\theta}}^n} = \cos{n\theta} \\
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& \Im{\paren{\cos{\theta} + i\sin{\theta}}^n} = \sin{n\theta} \\
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\end{align*}
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Binomial Expansion by Pascal's Triangle - sum of powers of the term are the number in pascal's triangle \\
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1, sum of two above
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\section*{Power Series}
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\begin{align*}
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\forall x \in \R .& ~ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} \\
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& \cos{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n}}{\paren{2n}!}}
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& \sin{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n+1}}{\paren{2n+1}!}}
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\end{align*}
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Define these for $x \in \C$ in the same way
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\begin{align*}
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\forall b \in \R . & ~ e^{bi} = 1 + bi + \frac{(bi)^2}{2!} + \frac{(bi)^3}{3!} + ...\\
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& = \paren{1 - \frac{b^2}{2!} + \frac{b^4}{4!}} + i\paren{b - \frac{b^3}{3!} + \frac{b^5}{5!}} \\
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& = \cos{b} + i\sin{b} \\
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\therefore \\
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& e^{a + bi} = e^a\cdot e^{bi} = \polar{e^a}{b}
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\end{align*}
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\subsection*{Consequences}
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$$ e^{i\pi} = -1 $$
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De Moivre's Theorem: $$ e^{i\alpha}e^{i\beta} = e^{i\paren{\alpha+\beta}} $$
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\subsection*{$\sin$, $\cos$ of Complex Numbers}
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\begin{align*}
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\forall \theta \in \R. \quad & e^{i\theta} + e^{-i\theta} = \cos{\theta} + i\sin{\theta} + \cos{-\theta} + i\sin{-\theta} \\
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& = \cos{\theta} + i\sin{\theta} + \cos{\theta} - i\sin{\theta} \\
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& = 2\cos{\theta} \\
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\therefore \\
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& \cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2} \quad \forall \theta \in \R
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\end{align*}
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Similarly, \[ \sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2i}\quad\forall\theta\in\R \] \\
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Define $\cos{\theta}$ as \[ \frac{e^{i\theta} + e^{-i\theta}}{2}$ $~\forall \theta \in \C \]
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Define $\sin{\theta}$ as \[ \frac{e^{i\theta} - e^{-i\theta}}{2i}$ $~\forall \theta \in \C \]
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\subsubsection*{Examples}
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\begin{align*}
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\cos{\paren{1 + i}} = \frac{e^{i\paren{1 + i}} + e^{-i\paren{1 + i}}}{2} = \frac{e^{i - 1} + e^{1 - i}}{2}
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= \frac{1}{2}\paren{\polar{e^{-1}}{1} + \polar{e}{-1}}
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\end{align*}
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\subsection*{Logarithms of Complex Numbers}
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$$ \forall y \in \R, y > 0. \quad e^{\ln{y}} = y $$
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\begin{align*}
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\forall z \in \C. \quad & e^{\ln{z}} = z \\
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& \text{let } w = \ln{z} = c + di \\
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& e^w = e^{c + di} = \polar{e^c}{d} = z \\
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& \implies e^c = \mod{z}, d = \arg{z} \\
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& \implies w = c + di = \ln{\mod{z}} + i\arg{z}
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\end{align*}
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Note: $\arg{z}$ is multivalued, therefore $\ln{z}$ is multivalued
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$$ \ln{z} = \ln{\mod{z}} + i\arg{z} $$
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Sidenote: $\forall a \in \R, a < 0. \quad \ln{a} = \ln{\abs{a}} + i\pi\paren{2n + 1} \forall n \in \N$
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$$ \forall z, w \in \C. \quad z^w = e^{w\ln{z}} $$
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Exponents of complex numbers are \emph{odd}. $\quad z^{w_1\cdot w_2} \leq \paren{z^{w_1}}^{w_2}$ - They aren't necessarily equal
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\end{document}
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58
MA1006/decls.tex
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MA1006/decls.tex
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\documentclass[fleqn]{article}
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\usepackage{amsmath,amssymb}
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\usepackage[margin=0.25in]{geometry}
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\usepackage{enumitem}
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\usepackage{systeme}
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\usepackage{mathtools}
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\date{}
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\author{}
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\renewcommand{\Re}[1]{\operatorname{\mathbb{R}e}(#1)}
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\renewcommand{\Im}[1]{\operatorname{\mathbb{{I}}m}(#1)}
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\newcommand{\C}{\mathbb{C}}
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\newcommand{\N}{\mathbb{N}}
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\newcommand{\Z}{\mathbb{Z}}
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\newcommand{\Q}{\mathbb{Q}}
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\newcommand{\R}{\mathbb{R}}
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\newcommand{\conj}[1]{\overline{#1}}
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\renewcommand{\mod}[1]{\left|#1\right|}
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\newcommand{\abs}[1]{\left|#1\right|}
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\newcommand{\paren}[1]{\left(#1\right)}
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\newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}}
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\makeatletter
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\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
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\hskip -\arraycolsep
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\let\@ifnextchar\new@ifnextchar
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\array{#1}}
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\makeatother
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% https://gitlab.com/jim.hefferon/linear-algebra/-/blob/master/src/sty/linalgjh.sty
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\newlength{\grsteplength}
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\setlength{\grsteplength}{1.5ex plus .1ex minus .1ex}
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\newcommand{\grstep}[2][\relax]{%
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\ensuremath{\mathrel{
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\hspace{\grsteplength}\mathop{\longrightarrow}\limits^{#2\mathstrut}_{
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\begin{subarray}{l} #1 \end{subarray}}\hspace{\grsteplength}}}}
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\newcommand{\repeatedgrstep}[2][\relax]{\hspace{-\grsteplength}\grstep[#1]{#2}}
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\newcommand{\swap}{\leftrightarrow}
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% https://tex.stackexchange.com/a/198806
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\makeatletter
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\newcommand{\subalign}[1]{%
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\vcenter{%
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\Let@ \restore@math@cr \default@tag
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\baselineskip\fontdimen10 \scriptfont\tw@
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\advance\baselineskip\fontdimen12 \scriptfont\tw@
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\lineskip\thr@@\fontdimen8 \scriptfont\thr@@
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\lineskiplimit\lineskip
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\ialign{\hfil$\m@th\scriptstyle##$&$\m@th\scriptstyle{}##$\hfil\crcr
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#1\crcr
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}%
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}%
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}
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\makeatother
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243
MA1006/linear.tex
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MA1006/linear.tex
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\input{decls.tex}
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\title{Linear Algebra}
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\begin{document}
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\maketitle
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\section*{Allowable Operations on a Linear System}
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Solutions invariant.
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\begin{itemize}
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\item Multiply an equation by a non-zero scalar
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\item Swap two equations
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\item Add a multiple of one equation to another
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\end{itemize}
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\subsection*{Example}
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\begin{align*}
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&\systeme{
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x - 2y + 2z = 6,
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-x + 3y + 4z = 2,
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2x + y - 2z = -2
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}\\\\
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E_2 & \implies E_2 + E_1 \\
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E_3 & \implies E_3 + E_1 \\
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&\systeme{
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x - 2y + 2z = 6,
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y + 6z = 8,
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5y - 6z = -14
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}\\\\
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E_3 & \implies E_3 - 5E_2 \\
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&\systeme{
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x - 2y + 2z = 6,
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y + 6z = 8,
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z = \frac{3}{2}
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}\\\\
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E_1 & \implies E_1 - 2E_3 \\
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E_2 & \implies E_2 - 6E_3 \\
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&\systeme{
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x - 2y = 3,
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y = -1,
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z = \frac{3}{2}
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}\\\\
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E_1 & \implies E_1 + 2E_2 \\
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&\systeme{
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x = 1,
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y = -1,
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z = \frac{3}{2}
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}\\\\
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\end{align*}
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\section*{As Matrices}
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\begin{align*}
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\systeme{
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x + 2y = 1,
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2x - y = 3
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}
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\quad=\quad
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\begin{pmatrix}[cc|c]
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1 & 2 & 1 \\
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2 & -1 & 3
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\end{pmatrix}
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& \systeme{
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x - y + z = -2,
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2x + 3y + z = 7,
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x - 2y - z = -2
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} \quad=\quad \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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2 & 3 & 1 & 7 \\
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1 & -2 & -1 & -2
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\end{pmatrix} \\
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\grstep[R_3 - R_1]{R_2 - 2R_1} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & -1 & -2 & 0
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\end{pmatrix} \\
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\grstep{5R_3 + R_2} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & 0 & -11 & 11 \\
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\end{pmatrix} \\
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\grstep{-11^{-1}R_3} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\
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\grstep[R_1 - R_3]{R_2 + R_3} & \begin{pmatrix}[ccc|c]
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1 & -1 & 0 & -1 \\
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0 & 5 & 0 & 10 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\&
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\grstep{5^{-1}R_2} & \begin{pmatrix}[ccc|c]
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1 & -1 & 0 & -1 \\
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0 & 1 & 0 & 2 \\
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0 & 0 & 1 & -1 \\
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\end{pmatrix} \\
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\grstep{R_1 + R_2} & \begin{pmatrix}[ccc|c]
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1 & 0 & 0 & 1 \\
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0 & 1 & 0 & 2 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\
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= & \quad
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\left\{
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\subalign{
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x & ~= ~1 \\
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y & ~= ~2 \\
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z & ~= ~-1
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||||
}
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\right.
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\end{align*}
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||||
\section*{Row-Echelon Form}
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\begin{description}
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||||
\item[Row-Echelon Form] The leading entry in each row is 1 and is further to the right than the previous row's leading entry,
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all 0 rows are at the end
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\item[Reduced Row-Echelon Form] every other entry in a column containing a leading 1 is 0
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||||
\item[Theorem:] A matrix can be transformed to reduced row-echelon form using a finite number of allowable row operations
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||||
\end{description}
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||||
\subsection*{Example}
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\begin{align*}
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& \systeme{3x_1 + 2x_2 = 1,
|
||||
x_1 - x_2 = 4,
|
||||
2x_1 + x_2 = 5} = \begin{pmatrix}[cc|c]
|
||||
3 & 2 & 1 \\
|
||||
1 & -1 & 4 \\
|
||||
2 & 1 & 5
|
||||
\end{pmatrix} \\
|
||||
\grstep{R_1\swap R_2} & \begin{pmatrix}[cc|c]
|
||||
1 & -1 & 4 \\
|
||||
3 & 2 & 1 \\
|
||||
2 & 1 & 5
|
||||
\end{pmatrix} \\
|
||||
\grstep[R_2 - 3R_1]{R_3 - 2R_1} & \begin{pmatrix}[cc|c]
|
||||
1 & -1 & 4 \\
|
||||
0 & 5 & -11 \\
|
||||
0 & 3 & -3
|
||||
\end{pmatrix} \\
|
||||
\grstep{5^{-1}R_2} & \begin{pmatrix}[cc|c]
|
||||
1 & -1 & 4 \\
|
||||
0 & 1 & \frac{-11}{5} \\
|
||||
0 & 3 & -3
|
||||
\end{pmatrix} \\
|
||||
\grstep{R_3 - 2R_2} & \begin{pmatrix}[cc|c]
|
||||
1 & -1 & 4 \\
|
||||
0 & 1 & \frac{-11}{5} \\
|
||||
0 & 0 & \frac{18}{5}
|
||||
\end{pmatrix} \\
|
||||
= & \systeme{
|
||||
x_1 - x_2 = 4,
|
||||
x_2 = \frac{-11}{5},
|
||||
0x_1 + 0x_2 = \frac{18}{5}
|
||||
}
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
& \begin{pmatrix}[cccc|c]
|
||||
1 & -1 & 1 & 1 & 6 \\
|
||||
-1 & 1 & -2 & 1 & 3 \\
|
||||
2 & 0 & 1 & 4 & 1 \\
|
||||
\end{pmatrix} \\
|
||||
\grstep[R_2 + R_1]{R_3 - 2R_1} & \begin{pmatrix}[cccc|c]
|
||||
1 & -1 & 1 & 1 & 6 \\
|
||||
0 & 0 & -1 & 2 & 9 \\
|
||||
0 & 2 & -1 & 2 & -11
|
||||
\end{pmatrix} \\
|
||||
\grstep[R_2\swap R_3]{2^{-1}R_3} & \begin{pmatrix}[cccc|c]
|
||||
1 & -1 & 1 & 1 & 6 \\
|
||||
0 & 1 & \frac{1}{2} & 1 & \frac{-11}{2} \\
|
||||
0 & 0 & -1 & 2 & 9 \\
|
||||
\end{pmatrix} \\
|
||||
\grstep[R_1 + R_3]{R_2 - 2^{-1}R_3} & \begin{pmatrix}[cccc|c]
|
||||
1 & -1 & 0 & 3 & 15 \\
|
||||
0 & 1 & 0 & 0 & -10 \\
|
||||
0 & 0 & -1 & 2 & 9 \\
|
||||
\end{pmatrix} \\
|
||||
\grstep[-R_3]{R_1 + R_2} & \begin{pmatrix}[cccc|c]
|
||||
1 & 0 & 0 & 3 & 15 \\
|
||||
0 & 1 & 0 & 0 & -10 \\
|
||||
0 & 0 & 1 & -2 & -9 \\
|
||||
\end{pmatrix} \\
|
||||
= & \systeme{
|
||||
x_1 + 3x_4 = 5,
|
||||
x_2 = -10,
|
||||
x_3 - 2x_4 = -9
|
||||
} \\
|
||||
= & \left\{\substack{
|
||||
x_1 = 5 - 3t \\
|
||||
x_2 = -10 \\
|
||||
x_3 = -9 + 2t
|
||||
}\right.
|
||||
\end{align*}
|
||||
\section*{Determinants}
|
||||
The determinant of a matrix is defined only for square matrices.
|
||||
\[\det{A} \neq 0 \iff \exists \text{ a unique solution to the linear system represented by } A\]
|
||||
Let
|
||||
\[A = \begin{pmatrix}
|
||||
a_{11} & a_{12} & a_{1n} \\
|
||||
a_{21} & \ddots & \vdots \\
|
||||
a_{31} & \ldots & a_{3n} \\
|
||||
\end{pmatrix}
|
||||
\]
|
||||
\begin{description}
|
||||
\item[$i, j$ minor of $A$] an $n$x$n$ matrix constructed by removing the $i^\text{th}$ row and $j^\text{th}$ column of $A$ \\
|
||||
Denoted by $A_{ij}$
|
||||
\end{description}
|
||||
\begin{align*}
|
||||
& \det{A} \text{ where } n = 1. = a_{11} \\
|
||||
& \det{A} = a_{11}\det{A_{11}} - a_{12}\det{A_{12}} + ... + (-1)^{n+1}a_{1n} \tag{Laplace expansion of the first row} \\
|
||||
& \qquad \text{or laplace expansion along other row or column}
|
||||
\text{For } n = 2:& \\
|
||||
& \det{A} = a_{11}\cdot a_{22} - a_{12}\cdot a_{21}
|
||||
\end{align*}
|
||||
\begin{description}
|
||||
\item[Upper Triangular] lower left triangle is 0 - $d_{ij} = 0 \quad \forall{i > j}$
|
||||
\item[Lower Triangular] upper right triangle is 0 - $d_{ij} = 0 \quad \forall{i < j}$
|
||||
\item[Diagonal] only values on the diagonal - $d_{ij} = 0 \quad \forall{i \neq j}$ \\
|
||||
$\det{A} = \prod^{N}_{i=0}~a_{ij} \forall~\text{ row-echelon }A$
|
||||
\end{description}
|
||||
\begin{itemize}
|
||||
\item Multiplying a row of a square matrix $A$ by $r$ multiplies $\det{A}$ by $r$
|
||||
\item Swapping two rows of a square matrix $A$ multiplies $\det{A}$ by $-1$
|
||||
\item Adding a multiple of a row does not effect the determinant
|
||||
\end{itemize}
|
||||
\section*{Transposition}
|
||||
\begin{description}
|
||||
\item[$A^T$] $a^T_{ij} = a_{ji}~ \forall~i,j$
|
||||
\end{description}
|
||||
Note: $\det{A} = \det{A^T}~\forall~A$
|
||||
\section*{Matrix Multiplication}
|
||||
LHS has columns $=$ rows of RHS
|
||||
It's the cartesian product
|
||||
\[A\times B = (a_{i1}b_{j1} + a_{i2}b_{2j} + \ldots + a_{im}b_{mj})_{ij}\]
|
||||
\begin{align*}
|
||||
\begin{pmatrix}[c|c|c]
|
||||
2 & 1 + 1 & 3 + 6 \\
|
||||
4(2) & 4 + 1 & 3(4) + 6 \\
|
||||
0 & 2 & 2(6) \\
|
||||
\end{pmatrix} = \begin{pmatrix}
|
||||
2 & 2 & 9 \\
|
||||
8 & 5 & 18 \\
|
||||
0 & 2 & 12
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}\begin{pmatrix}1 & 2 & 3 & 4\end{pmatrix} + \begin{pmatrix}
|
||||
1 & 2 & 3 & 4 \\
|
||||
5 & 6 & 7 & 8 \\
|
||||
9 & 10 & 11 & 12 \\
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
\end{document}
|
||||
9
MA1006/notes-2023-09-21
Normal file
9
MA1006/notes-2023-09-21
Normal file
@@ -0,0 +1,9 @@
|
||||
Leading coefficient of a polynomial is the coefficient of the highest power of x.
|
||||
Leading coefficient is inherently non-zero
|
||||
Monic polynomials: leading coefficient == 1
|
||||
|
||||
Rational Root Theorem:
|
||||
forall P(x): polynomial of degree n with integer coefficients
|
||||
forall p, q: Z such that p/q is a root of P(x)
|
||||
p is a factor of a[0]
|
||||
q is a factor of a[n]
|
||||
34
MA1006/polynomials.tex
Normal file
34
MA1006/polynomials.tex
Normal file
@@ -0,0 +1,34 @@
|
||||
\input{decls.tex}
|
||||
\title{Polynomials}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\begin{itemize}[leftmargin=10em]
|
||||
\item[Polynomial in $\C$] \quad $\forall n \in \N, a_{i \in [0, n]} \in \C, a_n \neq 0. \quad P(x) = \sum_{i=0}^{n} a_{i}x^i$
|
||||
\item[Degree] \quad $n$
|
||||
\item[Leading Coefficient] \quad $a_n$
|
||||
\item[Monic Polynomial] \quad Polynomial with $a_n = 1$
|
||||
\end{itemize}
|
||||
The Abel-Ruffini theorem states that there exists a degree-5 polynomial with roots that cannot be expressed with $+ - * / \surd$
|
||||
\section*{Rational Root Theorem}
|
||||
\[
|
||||
\forall \text{ polynomials } P(x) \text { with } \forall i. a_i \in \Z.\quad \forall \frac{p}{q} \in \Q.~P\paren{\frac{p}{q}} = 0 \implies p|a_0 \land q|a_n \\ % x|y x divides y
|
||||
\]
|
||||
This means that monic polynomials have no rational non-integral roots.
|
||||
\section*{Polynomial Division}
|
||||
\[\forall P(x), D(x).~ \exists Q(x), R(x).~ P(x) = D(x)Q(x) + R(x), \operatorname{degree}(R) < \operatorname{degree}(D)\]
|
||||
\section*{Remainder Theorem}
|
||||
\[\forall P(x), c. P(c) = 0 \iff \paren{x - c}|P(x)\]
|
||||
Proof:
|
||||
\begin{align*}
|
||||
\text{Given that }& P(c) = 0: \\
|
||||
& \exists Q(x), R(x).~P(x) = Q(x)(x - c) + R(x) & \text{Division of polynomials} \\
|
||||
& P(c) = Q(c)(c - c) + R(c) & \\
|
||||
& 0 = Q(c)(0) + R(c) & \\
|
||||
& R(c) = 0 & \\
|
||||
& \forall x.~R(x) = 0 & \text{As $D(x)$ has degree $1$, $R(x)$ must have degree $0$} \\
|
||||
& \therefore~(x - c)|P(x) \\
|
||||
\text{Given that }& (x - c)|P(x) \\
|
||||
& \exists Q(x).~P(x) = Q(x)(x - c) & \text{Division of polynomials, remainder $0$} \\
|
||||
& \therefore~P(c) = Q(c)(c - c) = 0\cdot Q(c) = 0
|
||||
\end{align*}
|
||||
\end{document}
|
||||
6
MA1006/tmpl.tex
Normal file
6
MA1006/tmpl.tex
Normal file
@@ -0,0 +1,6 @@
|
||||
\input{decls.tex}
|
||||
\title{}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\end{document}
|
||||
42
MA1006/tut-2023-10-10.tex
Normal file
42
MA1006/tut-2023-10-10.tex
Normal file
@@ -0,0 +1,42 @@
|
||||
\input{decls.tex}
|
||||
\title{Tutorial 2023-10-10}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\section{}
|
||||
\section{}
|
||||
They're rotated by 90 degrees
|
||||
\section{}
|
||||
\begin{tabular}{ c c c c }
|
||||
$\theta$ & $\cos\theta$ & $\sin\theta$ & $\tan\theta$ \\ \hline
|
||||
$0$ & $ 1 $ & $ 0 $ & $ 0 $ \\ \hline
|
||||
$\pi/6$ & $\sqrt{3}/2$ & $1/2$ & $\sqrt{3}/3$ \\ \hline
|
||||
$\pi/4$ & $\sqrt{2}/2$ & $\sqrt{2}/2$ & $1$ \\ \hline
|
||||
$\pi/3$ & $1/2$ & $\sqrt{3}/2$ & $\sqrt{3}$ \\ \hline
|
||||
$\pi/2$ & $0$ & $1$ & undefined \\ \hline
|
||||
$-\pi/6$ & $\sqrt{3}/2$ & $-1/2$ & $-\sqrt{3}/3$ \\ \hline
|
||||
$-\pi/4$ & $\sqrt{2}/2$ & $-\sqrt{2}/2$ & -1 \\ \hline
|
||||
$-\pi/3$ & $1/2$ & $-\sqrt{3}/2$ & $-\sqrt{3}$ \\ \hline
|
||||
$-\pi/2$ & $0$ & $-1$ & undefined \\ \hline
|
||||
\end{tabular}
|
||||
\section{}
|
||||
\subsection{}
|
||||
\begin{align*}
|
||||
& +1 = \polar{1}{0} & {-1} = \polar{1}{\pi} \\
|
||||
& +i = \polar{1}{\frac{\pi}{2}} & {-i} = \polar{1}{-\frac{\pi}{2}}
|
||||
\end{align*}
|
||||
\subsection{}
|
||||
\[1 - i = \polar{\sqrt{2}}{\frac{-\pi}{4}}\]
|
||||
\[1 + i = \polar{\sqrt{2}}{\frac{\pi}{4}} \]
|
||||
\subsection{}
|
||||
\[-1 + i\sqrt{3} = \polar{2}{\frac{-\pi}{3}}\]
|
||||
\[-1 - i\sqrt{3} = \polar{2}{\frac{+\pi}{3}}\]
|
||||
\subsection{}
|
||||
\[\]
|
||||
\section{}
|
||||
\subsection{}
|
||||
\[i\]
|
||||
\subsection{}
|
||||
\[\sqrt{2} + \sqrt{2}i\]
|
||||
\subsection{}
|
||||
|
||||
\end{document}
|
||||
9
MA1006/tut-2023-10-17.tex
Normal file
9
MA1006/tut-2023-10-17.tex
Normal file
@@ -0,0 +1,9 @@
|
||||
\input{decls.tex}
|
||||
\title{Tutorial}
|
||||
\date{2023-10-17}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\section{}
|
||||
\subsection{}
|
||||
\begin{align*}
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user