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MATH 115: Linear Algebra
Set theory
!!! definition - Natural numbers (\(\mathbb N\)) are all integers greater than zero. - Integers (\(\mathbb Z\)) are all non-decimal numbers. - Rational numbers (\(\mathbb Q\)) are all numbers representable as a fraction. - Irrational numbers are all real numbers not representable as a fraction. - Real numbers (\(\mathbb R\)) are all rational or irrational numbers.
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.
!!! example - Natural numbers are a subset of integers, or \(\mathbb N \subseteq \mathbb Z\). - Integers are not a subset of natural numbers, or \(\mathbb Z \not\subseteq \mathbb N\).
!!! warning 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.
Sets can be subtracted with a backslash (\), returning a set with all elements in the first set not in the second.
!!! example The set of irrational numbers can be represented as the difference between the real and rational number sets, or: \[\mathbb R \backslash \mathbb Q\]
Complex numbers
A complex number can be represented in the form: \[x+yj\]
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.
!!! definition - \(Re(z)\) is the real component of complex number \(z\). - \(Im(z)\) is the imaginary component of complex number \(z\).
These numbers can be treated effectively like any other number.
Properties of complex numbers
All of these properties can be derived from expanding the standard forms.
Where \(z=x+yj\) and \(w=a+bj\):
\[ \begin{align*} zw&=(ax-by)+(bx+ay)j \\ \frac{1}{z} &= \frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}j \\ z^0 &= 1 \end{align*} \]
??? example If \(z=2+5j\) and \(w=1+3j\): \[ \begin{align*} \frac{z}{w} &= (2+5j)(\frac{1}{1+9}-\frac{3}{1+9}j) \\ &= (2+5j)(\frac{1}{10}-\frac{3}{10j}) \\ &= \frac{1}{5}-\frac{3}{5}j+\frac{1}{2}j+\frac{3}{2} \\ &= \frac{17}{10}-\frac{1}{10}j \end{align*} \]
??? example To solve for \(z\) in \(z^2+4=0\): \[ \begin{align*} (x+yj)^2&=-4 \\ x^2+2xyj - y^2 &= -4 + 0j \\ (x^2-y^2) + 2xyj &= -4+0j \\ \\ ∵ x, y \in \mathbb R: 2xyj &= 0j \\ ∴ \begin{cases} x^2-y^2=-4 \\ 2xy = 0 \end{cases} \\ \\ x=0 &\text{ or } y=0 \\ \pu{if } x=0&: y =\pm 2 \\ \pu{if } y=0&: \text{no real solutions} \\ \\ ∴ x&=0, y=\pm 2 \\ z&=\pm 2j \end{align*} \]
??? example To solve for \(z\) in \(z^2=5+12j\): \[ \begin{align*} (x+yj)^2&=5+12j \\ (x^2-j^2)+2xyj = 5+12j \\ \\ \begin{cases} x^2-y^2=5 \\ 2xy = 12 \end{cases} \\ \\ y &= \frac{6}{x} \\ x^2 - \frac{6}{x}^2 &= 5\\ x^4 - 36 - 5x^2 &= 0 \\ x^2 &= 9, -4, x\in \mathbb R \\ x &= 3, -3 \\ y &= 2, -2 \\ z &= 3+2j, -3-2j \end{align*} \]
Conjugates
The conjugate of any number can be written with a bar above it. \[\overline{x+yj} = x-yj\]
The conjugate of a conjugate is the original number. \[\overline{\overline{ z}} = z\]
\(z\) is a real number if and only if its conjugate is itself. \[z\in\mathbb R \iff \overline{z}=z\]
\(z\) is purely imaginary if and only if its conjugate is the negative version of itself. \[z\in\text{only imaginary} \iff \overline{z}=-z\]
Conjugates are flexible and can almost be treated as just another factor. \[ \begin{align*} \overline{z+w}&=\overline{z}+\overline{w} \\ \overline{zw}&=(\overline{z})(\overline{w}) \\ \overline{z^k}&=\overline{z}^k \\ \overline{\biggr(\frac{z}{w}\biggr)} &= \frac{\overline{z}}{\overline{w}}, w\neq 0 \end{align*} \]
Modulus
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. \[|z| = \sqrt{x^2+y^2}\]
!!! example The modulus of complex number \(2-j\) is: \[ \begin{align*} |2-j|&=\sqrt{2^2+(-1)^2} \\ &= -5 \end{align*} \]
If there is no imaginary component, a complex number’s modulus is its absolute value. \[z\in\mathbb R: |z|=|Re(z)|\]
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.
Properties of moduli
These can be also be manually derived.
If the modulus is zero, the complex number is zero. \[|z|=0 \iff z=0\]
The modulus of the conjugate is equal to the modulus of the original. \[|\overline{z}| = |z|\]
The number multiplied by the conjugate modulus is the square of the modulus. \[z|\overline{z}|=|z|^2\]
Moduli are also almost just a factor: \[ \begin{align*} \biggr|\frac{z}{w}\biggr| &= \frac{|z|}{|w|}, w \neq 0 \\ |zw| &= |z||w| \end{align*} \]
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.
\[|z+w| \leq |z|+|w|\]
Geometry
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.
(Source:
Wikimedia Commons, GNU FGL 1.2 or later)
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.
Polar form
The variable \(r\) is equal to the modulus of a complex number \(|z|\).
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: \[z=r(\cos\theta + j\sin\theta)\]
Trigonometry can be used to calculate \(\cos\theta\) and \(\sin\theta\) as \(\cos\theta = \frac{x}{r}\) and \(\sin\theta = \frac{y}{r}\).
!!! example \(1+\sqrt{3}j=2\big(\cos\frac{\pi}{3} + j\sin\frac{\pi}{3}\big)\)
!!! warning The polar form is not unique because going around 360° results in the same vector. Where \(k\) is any integer. \[r(\cos\theta + j\sin\theta) = r(\cos(\theta+2k\pi) + j\sin(\theta+2k\pi))\]
The polar form is useful for the multiplication of complex numbers.
Because of the angle sum identities: \[z_1z_2=r_1r_2(\cos(\theta_1+\theta_2) + j\sin(\theta_1+\theta_2))\]
This can be extrapolated into Moivre’s theorem: \[z^n=r^n(\cos(n\theta) + j\sin(n\theta))\]
To determine the roots of a complex number, Moivre’s theorem can be used again: \[\sqrt[n]{z} = \sqrt[n]{r}\big(\cos\big(\frac{\theta + 2k\pi}{n}\big) + j\sin\big(\frac{\theta + 2k\pi}{n}\big)\big)\]
where \(k\) is every number in the range \([0, n-1], k\in\mathbb Z\).
!!! example To find all answers for \(w^5=-32\): \[ \begin{align*} w^5 &= 32(\cos\theta + \sin\theta) \\ w_k &= \sqrt[5]{32}\biggr[\cos\biggr(\frac{\pi + 2k\pi}{5}\biggr) + j\sin\biggr(\frac{\pi+2k\pi}{5}\biggr)\biggr] w_0 &= 2\biggr(\cos\frac{\pi}{5} + j\sin\frac{\pi}{5}\biggr) = 2e^{j\frac{\pi}{5}} \\ w_1 &= 2\biggr(\cos\frac{3\pi}{5} + j\sin\frac{3\pi}{5}\biggr) = 2e^{j\frac{3\pi}{5}} \\ w_2 &= 2(\cos\pi + j\sin\pi = 2e^{j\pi} \\ \\ \text{etc.} \end{align*} \]
The exponential form of a complex number employs Euler’s identity: \[ \begin{align*} e^{j\pi} &= -1 \\ e^{j\pi} &= \cos\theta + j\sin\theta \\ z &= re^{j\pi} \end{align*} \]
Proofs
!!! example
Vectors
Please see SL Math - Analysis and Approaches 2#Vectors and SL Physics 1#1.3 - Vectors and Scalars for more information.
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.
\[ \newcommand\colv[1]{\begin{bmatrix}#1\end{bmatrix}} \colv{x_1 \\ x_2 \\ x_3} \]
The zero vector is full of zeroes. \[ \colv{0 \\ 0 \\ 0} \]
!!! warning Vectors of different dimensions cannot be compared — the missing dimensions cannot be treated as 0.
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.