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# MATH 115: Linear Algebra
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## Set theory
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!!! definition
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- Natural numbers ($\mathbb N$) are all **integers** greater than zero.
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- Integers ($\mathbb Z$) are all non-decimal numbers.
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- Rational numbers ($\mathbb Q$) are all numbers representable as a fraction.
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- Irrational numbers are all **real** numbers not representable as a fraction.
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- Real numbers ($\mathbb R$) are all rational or irrational numbers.
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The **subset sign** ($\subseteq$) indicates that one **set** is strictly within another. The **not subset sign** ($\not\subseteq$) indicates that at least one element in the first set is not in the second.
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!!! example
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- Natural numbers are a subset of integers, or $\mathbb N \subseteq \mathbb Z$.
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- Integers are not a subset of natural numbers, or $\mathbb Z \not\subseteq \mathbb N$.
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!!! warning
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The subset sign is not to be confused with the **element of** sign ($\in$), as the former only applies to sets while the latter only applies to elements.
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Sets can be subtracted with a **backslash** (\\), returning a set with all elements in the first set not in the second.
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!!! example
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The set of irrational numbers can be represented as the difference between the real and rational number sets, or:
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$$\mathbb R \backslash \mathbb Q$$
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## Complex numbers
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A complex number can be represented in the form:
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$$x+yj$$
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where $x$ and $y$ are real numbers, and $j$ is the imaginary $\sqrt{-1}$ (also known as $i$ outside of engineering). This implies that every real number is also in the set of complex numbers as $y$ can be set to zero.
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!!! definition
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- $Re(z)$ is the **real component** of complex number $z$.
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- $Im(z)$ is the **imaginary component** of complex number $z$.
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These numbers can be treated effectively like any other number.
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### Properties of complex numbers
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All of these properties can be derived from expanding the standard forms.
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Where $z=x+yj$ and $w=a+bj$:
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$$
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\begin{align*}
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zw&=(ax-by)+(bx+ay)j \\
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\frac{1}{z} &= \frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}j \\
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z^0 &= 1
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\end{align*}
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$$
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??? example
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If $z=2+5j$ and $w=1+3j$:
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$$
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\begin{align*}
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\frac{z}{w} &= (2+5j)(\frac{1}{1+9}-\frac{3}{1+9}j) \\
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&= (2+5j)(\frac{1}{10}-\frac{3}{10j}) \\
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&= \frac{1}{5}-\frac{3}{5}j+\frac{1}{2}j+\frac{3}{2} \\
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&= \frac{17}{10}-\frac{1}{10}j
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\end{align*}
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$$
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??? example
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To solve for $z$ in $z^2+4=0$:
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$$
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\begin{align*}
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(x+yj)^2&=-4 \\
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x^2+2xyj - y^2 &= -4 + 0j \\
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(x^2-y^2) + 2xyj &= -4+0j \\
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\\
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∵ x, y \in \mathbb R: 2xyj &= 0j \\
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∴ \begin{cases}
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x^2-y^2=-4 \\
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2xy = 0
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\end{cases} \\
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\\
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x=0 &\text{ or } y=0 \\
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\pu{if } x=0&: y =\pm 2 \\
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\pu{if } y=0&: \text{no real solutions} \\
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\\
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∴ x&=0, y=\pm 2 \\
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z&=\pm 2j
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\end{align*}
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$$
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??? example
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To solve for $z$ in $z^2=5+12j$:
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$$
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\begin{align*}
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(x+yj)^2&=5+12j \\
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(x^2-j^2)+2xyj = 5+12j \\
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\\
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\begin{cases}
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x^2-y^2=5 \\
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2xy = 12
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\end{cases} \\
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\\
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y &= \frac{6}{x} \\
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x^2 - \frac{6}{x}^2 &= 5\\
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x^4 - 36 - 5x^2 &= 0 \\
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x^2 &= 9, -4, x\in \mathbb R \\
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x &= 3, -3 \\
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y &= 2, -2 \\
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z &= 3+2j, -3-2j
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\end{align*}
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$$
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### Conjugates
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The conjugate of any number can be written with a bar above it.
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$$\overline{x+yj} = x-yj$$
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The conjugate of a conjugate is the original number.
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$$\overline{\overline{ z}} = z$$
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$z$ is a real number **if and only if** its conjugate is itself.
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$$z\in\mathbb R \iff \overline{z}=z$$
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$z$ is purely imaginary **if and only if** its conjugate is the negative version of itself.
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$$z\in\text{only imaginary} \iff \overline{z}=-z$$
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Conjugates are flexible and can almost be treated as just another factor.
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$$
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\begin{align*}
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\overline{z+w}&=\overline{z}+\overline{w} \\
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\overline{zw}&=(\overline{z})(\overline{w}) \\
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\overline{z^k}&=\overline{z}^k \\
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\overline{\biggr(\frac{z}{w}\biggr)} &= \frac{\overline{z}}{\overline{w}}, w\neq 0
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\end{align*}
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$$
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### Modulus
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The modulus of a number is represented by the absolute value sign. It is equal to its magnitude if the complex number were a vector.
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$$|z| = \sqrt{x^2+y^2}$$
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!!! example
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The modulus of complex number $2-j$ is:
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$$
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\begin{align*}
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|2-j|&=\sqrt{2^2+(-1)^2} \\
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&= -5
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\end{align*}
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$$
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If there is no imaginary component, a complex number's modulus is its absolute value.
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$$z\in\mathbb R: |z|=|Re(z)|$$
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Complex numbers cannot be directly compared because imaginary numbers have no inequalities, but their moduli can — the modulus of one complex number can be greater than another's.
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#### Properties of moduli
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These can be also be manually derived.
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If the modulus is zero, the complex number is zero.
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$$|z|=0 \iff z=0$$
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The modulus of the conjugate is equal to the modulus of the original.
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$$|\overline{z}| = |z|$$
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The number multiplied by the conjugate modulus is the square of the modulus.
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$$z|\overline{z}|=|z|^2$$
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Moduli are also almost just a factor:
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$$
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\begin{align*}
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\biggr|\frac{z}{w}\biggr| &= \frac{|z|}{|w|}, w \neq 0 \\
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|zw| &= |z||w|
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\end{align*}
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$$
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The moduli of the sum is always less than the sum of the moduli of the individual numbers — this is also known as the triangle inequality theorem.
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$$|z+w| \leq |z|+|w|$$
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### Geometry
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In setting the x- and y-axes to the imaginary and real components of a complex number, complex numbers can be represented almost as vectors.
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<img src="https://upload.wikimedia.org/wikipedia/commons/6/69/Complex_conjugate_picture.svg">(Source: Wikimedia Commons, GNU FGL 1.2 or later)</img>
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The complex number $x+yj$ will be on the point $(x, y)$, and the modulus is the magnitude of the vector. Complex number moduli can be compared graphically if their points lie within a drawn circle centred on the origin with a point on another vector.
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### Polar form
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The variable $r$ is equal to the modulus of a complex number $|z|$.
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From the Pythagorean theorem, the polar form of a complex number can be expressed using the angle of the modulus to the real axis. Where $\theta$ is the angle of the modulus to the real axis:
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$$z=r(\cos\theta + j\sin\theta)$$
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Trigonometry can be used to calculate $\cos\theta$ and $\sin\theta$ as $\cos\theta = \frac{x}{r}$ and $\sin\theta = \frac{y}{r}$.
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!!! example
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$1+\sqrt{3}j=2\big(\cos\frac{\pi}{3} + j\sin\frac{\pi}{3}\big)$
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!!! warning
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The polar form is not unique because going around 360° results in the same vector. Where $k$ is any integer.
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$$r(\cos\theta + j\sin\theta) = r(\cos(\theta+2k\pi) + j\sin(\theta+2k\pi))$$
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The polar form is useful for the multiplication of complex numbers.
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Because of the angle sum identities:
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$$z_1z_2=r_1r_2(\cos(\theta_1+\theta_2) + j\sin(\theta_1+\theta_2))$$
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This can be extrapolated into Moivre's theorem:
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$$z^n=r^n(\cos(n\theta) + j\sin(n\theta))$$
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To determine the roots of a complex number, Moivre's theorem can be used again:
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$$\sqrt[n]{z} = \sqrt[n]{r}\big(\cos\big(\frac{\theta + 2k\pi}{n}\big) + j\sin\big(\frac{\theta + 2k\pi}{n}\big)\big)$$
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where $k$ is every number in the range $[0, n-1], k\in\mathbb Z$.
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!!! example
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To find all answers for $w^5=-32$:
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$$
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\begin{align*}
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w^5 &= 32(\cos\theta + \sin\theta) \\
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w_k &= \sqrt[5]{32}\biggr[\cos\biggr(\frac{\pi + 2k\pi}{5}\biggr) + j\sin\biggr(\frac{\pi+2k\pi}{5}\biggr)\biggr]
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w_0 &= 2\biggr(\cos\frac{\pi}{5} + j\sin\frac{\pi}{5}\biggr) = 2e^{j\frac{\pi}{5}} \\
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w_1 &= 2\biggr(\cos\frac{3\pi}{5} + j\sin\frac{3\pi}{5}\biggr) = 2e^{j\frac{3\pi}{5}} \\
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w_2 &= 2(\cos\pi + j\sin\pi = 2e^{j\pi} \\
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\\
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\text{etc.}
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\end{align*}
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$$
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The **exponential** form of a complex number employs **Euler's identity**:
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$$
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\begin{align*}
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e^{j\pi} &= -1 \\
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e^{j\pi} &= \cos\theta + j\sin\theta \\
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z &= re^{j\pi}
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\end{align*}
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$$
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### Proofs
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!!! example
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## Vectors
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The column vector shows a vector of the form $(x, y, ...)$ from top to bottom as $(x_1, x_2, ...)$ as the number of dimensions increases.
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$$
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\newcommand\colv[1]{\begin{bmatrix}#1\end{bmatrix}}
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\colv{x_1 \\ x_2 \\ x_3}
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$$
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The zero vector is full of zeroes.
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$$
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\colv{0 \\ 0 \\ 0}
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$$
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!!! warning
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Vectors of different dimensions cannot be compared — the missing dimensions cannot be treated as 0.
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The standard form of a vector is written as the difference between two points: $\vec{OA}$ where $O$ is the origin and $A$ is any point. $\vec{AB}$ is the vector as a difference between two points.
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