# Test wiki wiki

#### Contents

## new stuff for linear alg

\[ \begin{bmatrix} 3 & 12 & 5 \\ 7 & 11 & 23 \\ 15 & -1 & 0 \end{bmatrix}\]

\[\theta\]

\[\hat{e_1}\] \[\hat{e_2}\]

\[ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}\]

\[O(n)\]

\[ \begin{bmatrix} A & B \\ C & D \end{bmatrix}\]

\[\frac{dH}{dv} = 0\]

\[ w^{(k+1)} = w^{(k)} + y_i x_i\]

\[P\left(A|B\right)\]

gradient descent:

\[\mathbf w_{j+1} = \mathbf w_j - \nu \nabla Q(\mathbf w_j)\]

vanishing gradient: \[f(x) = \text{max}(0,x) \]

\[y= m_1x^2 + m_2x + b\]

\[\lambda \cdot \sum_i m_i^2\]

LDA: \[P(X = x|k) = \frac {1} {\sqrt{2\pi\sigma} } e^{\frac{-1}{2\sigma^2}(x-\mu_k)^2}\]

KNN: *label this image with* \(K=6\)

Perceptron:

\[\vec{w}\cdot\vec{x} \ge b\]

Naive bayes:

\[P(k|x_1, x_2,\ldots,x_n)=\frac{P(x_1|k)P(x_2|k)\cdots P(x_n|k)P(k)}{P(x_1, x_2,\ldots,x_n)}\] or \[\frac{P(x_1|k)P(x_2|k)\cdots P(x_n|k)P(k)}{P(x_1, x_2,\ldots,x_n)}\] or \[P(k|x_1, x_2,\ldots,x_n)\]

Tree classification:

\[-\sum_{k=1}^{K} p_{mk}\log(p_{mk})\]

Boosting: \[f\left(\vec{x}\right) = \sum_{b=1}^{B}\lambda f^b\left(\vec{x}\right)\]

## old stuff

[Functions and Transformations: Composition] Text: f “circle” g (“circle” is composition operation) \[f \circ g\]

[Functions and Transformations: Inverses] Text: f^-1(x)

\[f^{-1}(x)\]

[Functions and Transformations: Inversion and Composition] Text: f^-1 “circle” f(x) = x

\[f^{-1}\circ f(x) = x\]

[Functions and Transformations: Recursion and Limits]
Text: x*{n+1} = f(x*{n})

\[x_{n+1} = f\left(x_n\right)\]

[Polynomials: Finding Roots] Description: Text: x^2-1=0 above x = plus or minus 1.

\[x^2-1=0\] \[x = \pm 1\]

[Polynomials: How Many Roots?]
Description: Part of a polynomial including **leading term with circle around power**.

\[x^3 + x = 1\]

[Polynomials: Factoring with Real Coefficients] Text: x^2-4x+3 = (x-1)(x-3)

\[x^2-4x+3 = (x-1)(x-3)\]

[Exponential Equations: Exponents Warmup] Text: 2^3 = ??

\[2^3 = \ ? \]

[Exponential Equations: Defining Exponents] Text: a^n = a X a X … X a

\[a^n = a \times a \times \cdots \times a\] \[a^n = a \cdot a \cdot \cdots \cdot a\]

[Exponential Equations: Laws of Exponents] Text: a^n X a^m =a^(m+n)

\[a^n \times a^m = a^{m+n}\] \[a^n \cdot a^m = a^{m+n}\]

[Logarithms: Understanding Log Arithmetic] Text: Log(ab) = Log(a) + Log(b).

\[\log(ab) = \log(a) + \log(b)\]

[Logarithms: Log Arithmetic Practice] Text: Log(a/b) = ??

\[\log \left(\frac{a}{b}\right) = \ ?\]

[Logarithms: Change of Base Formula]
Text: log*b (x) = log*a(x)/log_a(b)

\[\log_bx = \frac{\log_a x }{\log_a b}\]

[Logarithms: Log Equations] Text: ln (x^2+ 1) = 2

\[\ln \left(x^2+1\right) = 2\]

[Trigonometry: Relating the functions] Text: Tan(x) = Sin(x)/Cos(x)

\[\tan x =\frac{\sin x}{\cos x}\]

[Polar Coordinate: Simpler in Polar Form] Text: z = r e^{i theta}

\[z = re^{i \theta}\]

[Euler’s Formula: Just like normal algebra] Text: x^2 y^3/ x y^2 = x y

\[\frac{x^2 y^3}{xy^2} = xy\]

[Euler’s Formula: How to Approach Complex Exponents] Text: z^2 = …

\[z^2 = \cdots\] \[z^2 = \ldots\]

[Euler’s Formula: Adders] Text: +

\[+\]

[Euler’s Formula: Multipliers] Text: X \[\times \]

[Euler’s Formula: What is Exponentiation?] Text: e^x = … \[e^x = \cdots \] \[e^x = \ldots \]

[Euler’s Formula: Complex Exponentiation] Text: 2^i = … \[2^i=\cdots\] \[2^i=\ldots\]

[Limits: Indeterminate forms] Text: infinity/infinity \[\frac{\infty}{\infty}\]

[Derivatives: First Examples] Text: d/dx ( c ) = 0

\[\frac{d}{dx}(c) = 0 \]

[Derivatives: Second Derivative] Text: d^2 y /dx^2

\[\frac{d^2y}{dx^2}\]

[Computing Derivatives: Product and Quotient Rule] Text: (f g)’ = f’ g + g’ f (f/g)’ = (f g’ - g f’)/g^2 (should already exist in FIGMA)

\[\left(fg\right)' = f'g + g'f\] \[\left(f/g\right)' = \left(fg'-gf'\right)/g^2\]

[Computing Derivatives: Trig Functions] Text: d/dx [ sin(x) ] = cos(x)

\[\frac{d}{dx} \sin x = \cos x\]

\[\frac{d}{dx}\left(\sin x \right) = \cos x\]

[Computing Derivatives: Chain Rule] Text: dy/dx = dy/du du/dx

\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]

[Computing Derivatives: Exponentials and Logs] Text: d/dx (e^x) = e^x, d/dx [ ln(x) ] =1/x

\[\frac{d}{dx} e^x = e^x\] \[\frac{d}{dx} \ln x = \frac{1}{x}\]

[Linear Approximations and Applications: Implicit Differentiation] Text: d/dx ( x^2 + y^2 ) = …

\[\frac{d}{dx} \left(x^2 + y^2\right) = \cdots\] \[\frac{d}{dx} \left(x^2 + y^2\right) = \ldots\]

[Computing Integrals: Substitution] Text: int f(g(x)) g’(x) dx = int f(u) du

\[\int f\left(g(x)\right) g'(x)\,dx = \int f(u)\,du\]

[Computing Integrals: Exponentials and Trig] Text: int cos(x) dx = sin(x) + C \[\int \cos x \,dx = \sin x + C\] \[\int \cos(x) \,dx = \sin(x) + C\]

[Applications of Integrals: Work] Text: W = int F dx \[W = \int F \,dx\]

[First Order Equations: Separable] Text: dx/g(x) = dy/f(y)

\[ \frac{dx}{g(x)} = \frac{dy}{f(y)}\]

[First Order Equations: Homogeneous] Text: dy/dx + p(x) y = 0

\[\frac{dy}{dx} + p(x)y = 0\] \[\frac{dy}{dx} + p(x)\cdot y = 0\]

[First Order Equations: Nonhomogeneous] Text: dy/dx + p(x) y = q(x)

\[\frac{dy}{dx} + p(x)y = q(x)\] \[\frac{dy}{dx} + p(x)\cdot y = q(x)\]

[First Order Equations: Exact equations] Text: M dx + N dy = 0 \[M \,dx + N \,dy = 0\]

[Second Order Equations: Introduction] Text: d^2 h/dt^2 = - g

\[\frac{d^2h}{dt^2} = -g\]

[Vector Spaces: Intro to Vector Spaces] Text: v (with arrow on top) + 2 w (with arrow on top) (element symbol) V

\[\vec{v} + 2\vec{w} \in V\]

[Properties of Matrices: Matrix Algebra] Text: A B (not equals sign) B A \[AB \ne BA\]

[Properties of Matrices: INverses] Text: A^{-1} \[A^{-1}\]

[Linear Transformations: Linear Transformations] Text: T( x (with arrow on top) + c y (with arrow on top) ) = T( x (with arrow on top) ) + c T (y (with arrow on top) ) )

\[T\left(\vec{x} + c \vec{y}\right) = T\left(\vec{x}\right) + cT\left(\vec{y}\right)\]

[Linear Transformations: Properties of Transformations] Text: Nul(A) , Ker(A) \[\text{nul}(A)\] \[\ker(A)\] \[\text{nul } A\] \[\ker A\]

[Eigenvalues and Diagonalizability: Eigenvalues and Eigenvectors] Text: A x (with arrow on top) = lambda x (with arrow on top)

\[A\vec{x} = \lambda \vec{x}\]

[Eigenvalues and Diagonalizability: Characteristic Equation] Text: \[\text{det}\left(A-\lambda I\right) = 0\]

[Eigenvalues and Diagonalizability: Diagonalizability] Text: \[A = P D P^{-1}\]

[Introduction: Group Axioms] Text: \[(g_1\cdot g_2) \cdot g_3 = g_1 \cdot (g_2\cdot g_3)\]

[Fundamentals: Axioms] Text: g (belongs to) G arrow g^{-1} (belongs to) G

\[g \in G \Rightarrow g^{-1} \in G\]

[Fundamentals: Subgroups] Text: \[H \subset G\]

[Fundamentals: Abelian Groups] Text: \[AB = BA\]

[Fundamentals: Homomorphisms] Text: \[\phi : G \rightarrow H\]

[Fundamentals: Cosets] Text: \[G/H\]

[Advanced Topics: Normal Subgroups] Text: \[H \triangleleft G\]

[Advanced Topics: Isomorphism Theorems] Text: \[G/ \ker \phi \sim H\]

[Advanced Topics: Conjugacy Classes] Text: \[g H g^{-1}\]

[Advanced Topics: Symmetric Group] Text: \[S_{n}\]

[Advanced Topics: Signs of Permutations] Text: \[(12)(3)(456)\]

[Group Actions: Semidirect Products] Text: \[\text{Q}_8\]