awoo

Signed-off-by: Balazs Horvath <[email protected]>

LINEAR_ALGEBRA.md

@@ -6,15 +6,17 @@
 
 ---
 
-Linear Algebra is a branch of mathematics that deals with vectors, vector spaces and linear transformations.
+Linear Algebra is a branch of mathematics that deals with vectors, vector spaces and linear transformations. It has wide applications in fields like physics, computer graphics, data analysis and generating yiff.
 
 ### Vectors
 
 ---
 
-Vectors are entities that can be combined. They are often represented as an array of numbers, and each number corresponds to a coordinate in a space. A vector is a quantity that has both magnitude (or length) and direction often used to represent physical quantities such as force or velocity.
+Vectors are mathematical objects that have both magnitude (size) and direction. They can represent physical quantities like force, velocity or displacement. For example, if you're a furry running in a park, your velocity is a vector - it has a speed (magnitude) and a direction you're running in.
 
-Vectors can be represented in several ways, but one of the most common is an ordered list of numbers, known as a coordinate vector. For example vector $v$ might be represented as $v = [2, 3]$ which means it points 2 units in the `x` direction and 3 units in the `y` direction.
+Vectors are often represented as an array of numbers, where each number corresponds to a coordinate in space. In a 2D plane, a vector $v$ can be represented as an ordered pair: $v = [v_1, v_2]$. Here $v_1$ and $v_2$ are the components of $v$ along the x and y axes respectively.
+
+For example, the vector $v = [2, 3]$ can be visualized as an arrow starting at the origin $(0, 0)$ and pointing 2 units in the positive x-direction and 3 units in the positive y-direction.
 
 #### Vector Addition
 
@@ -32,41 +34,59 @@ The dot product of two vectors is a scalar quantity that is the sum of the produ
 
 The magnitude of a vector $g = [a, b]$ is given by the square root of the sum of the squares of its components. This is denoted as $||g||$ and calculated as $||g|| = \sqrt{a^2 + b^2}$
 
-
 ### Matrices
 
 ---
 
 A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent and manipulate linear equations. Each number in the matrix is called an element or entry. The position of an element is defined by its row number and column number.
 
+A `2x2` matrix can be represented as follows:
+
+$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$
+
 #### Matrix Addition and Subtraction
 
 Matrices can be added or subtracted element by element if they are of the same size. For example if we have two `2x2` matrices $A$ and $B$ then the sum $A+B$ is a new `2x2` matrix where each element is the sum of the corresponding elements in $A$ and $B$.
 
+It can be represented as:
+$$A + B = \begin{bmatrix} a1 & b1 \\ c1 & d1 \end{bmatrix} + \begin{bmatrix} a2 & b2 \\ c2 & d2 \end{bmatrix} = \begin{bmatrix} a1+a2 & b1+b2 \\ c1+c2 & d1+d2 \end{bmatrix}$$
+
 #### Scalar Multiplication of a Matrix
 
 A matrix can be multiplied by a scalar. This is done by multiplying each element of the matrix by the scalar.
 
+$$kA = k \cdot \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix}$$
+
 #### Matrix Multiplication
 
-The multiplication of two matrices is more complex than their addition. For two matrices to be multiplied, the number of columns in the first matrix must be qual to the number of rows in the second matrix.
+The multiplication of two matrices is more complex than their addition. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
+
+$$AB = \begin{bmatrix} a1 & b1 \\ c1 & d1 \end{bmatrix} \cdot \begin{bmatrix} a2 & b2 \\ c2 & d2 \end{bmatrix} = \begin{bmatrix} a1*a2 + b1*c2 & a1*b2 + b1*d2 \\ c1*a2 + d1*c2 & c1*b2 + d1*d2 \end{bmatrix}$$
 
 #### Identity Matrix
 
 This is a special type of square matrix where all the elements of the principal diagonal are ones and all other elements are zeros. The identity matrix play a similar role in matrix algebra as the number 1 in regular algebra.
 
+$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
+
 #### Determinant
 
 The determinant is a special number that can be calculated from a square matrix. It has many important properties and uses, such as providing the solution of a system of linear equations.
 
+$$det(A) = det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$
+
 #### Inverse
 
 The inverse of a matrix $A$ is another matrix, denoted as $A^{-1}$, such that when $A$ is multiplied by $A^{-1}$, the result is the identity matrix. Not all matrices have an inverse.
 
+$$A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$
+
 #### Transpose
 
 The transpose of a matrix is a new matrix whose rows are the columns of the original matrix and whose columns are the rows.
 
+$$A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$$
+
 - **Linear equations** are equations of the first order, representing straight lines in geometry. They are characterized by constants and variables without exponents or products of variables.
 - **Eigenvalues** and **Eigenvectors** are special sets of scalars and vectors associated with a matrix. They are fundamental in the study of linear transformations.
 - **Orthogonal Matrices** are square matrices whose columns and rows are orthogonal unit vectors (orthonormal vectors)

README.md

@@ -1295,7 +1295,7 @@ by spaceengine, a planet, black background
 - [⬇️ Download](https://huggingface.co/k4d3/yiff_toolkit/resolve/main/ponyxl_loras/blp-v1e400.safetensors?download=true)
 - [📊 Metadata](https://huggingface.co/k4d3/yiff_toolkit/raw/main/ponyxl_loras/blp-v1e400.json)
 
-Replicate [blp](https://e6ai.net/posts?tags=blp)'s unique style of AI art without employing 40 different custom nodes to alter sigmas and noise injection. I recommend you set your CFG to `6` and use `DPM++ 2M Karras` for the sampler and scheduler for a more realistic look or you can use `Euler a` for a more cartoony/dreamy generation with with a low CFG of `6`.
+Replicate [blp](https://e6ai.net/posts?tags=blp)'s unique style of AI art without employing 40 different custom nodes to alter sigmas and noise injection. I recommend you set your CFG to `6` and use `DPM++ 2M Karras` for the sampler and scheduler for a more realistic look or you can use `Euler a` for a more cartoony/dreamy generation with a low CFG of `6`.
 
 There have been reports that if you use this LoRA with a negative weight of `-0.5` your generations will have a slight sepia tone.