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Overview
Calculus
Computing Systems
Data
Graphs
Linear Algebra
Models and Modeling
Organization
Probability
Programming Constructs
- Programming Constructs Overview
- Abstraction
- Array
- Attribute
- Best-first Search
- Binary Search
- Block
- Branch
- Callback
- Class
- Conditional
- Constructor
- Container/Collection
- Dynamic Array
- Encapsulation
- Exception
- Expression
- Function
- Garbage Collection
- Greedy Algorithms
- Hash
- HTTP Request
- Identifier
- Inheritance
- Inner Class
- Instance
- Iterator
- Keyword
- Lambda
- Libraries
- List
- Linked List
- Literal
- Metaclass
- Method
- Mixin
- Object
- Operator
- Overloading
- Overriding
- Package
- Parameter
- Polymorphism
- Primitive
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- Recursion
- Reflection
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Linear Equations

Linear Equations are mathematical expressions for straight lines consisting of variables and coefficients.

A generalized form is:

$\\c_{1}v_{1}\,+\,\cdots\,+c_{n}v_{n}\,+\,b=0\\\\\textup{where:} \\\\v\,\,\,\,\,\,\textup{variables (unknowns), which can represent spatial dimensions} \\c\,\,\,\,\,\,\,\textup{coefficients for the variables, which can be considered parameters of the equation} \\b\,\,\,\,\,\,\,\textup{coefficient which can represent an offset of a vector from the origin}$

Linear Equations Applied to Multi-dimensional Spaces

Linear Equations can represent vectors in multi-dimensional spaces. In the diagram below, two vectors represented by black arrows show a spacial illustration of linear equation for (x,y) and (x,y,z):

Using the generalized equation from above and the sample data points from the diagram above, we can formulate linear equations for the two vectors:

Linear Equation for the (x,y) Vector

$\\\\\textup{using:} \\\\c_{1}=-\frac{1}{2} \\\\v_{1}=x\,axis\,value=-2 \\\\c_{2}=-1 \\\\v_{2}=y\,axis\,value=1 \\\\b=0 \\\\\\\textup{gives:} \\\\\left( -\frac{1}{2}\cdot -2 \right)+\left( -1\cdot 1\right )+0=0 \\\\\\\textup{yielding the balanced equation:} \\\\1-1=0$

Linear Equation for the (x,y,z) Vector

$\\\\\textup{using:} \\\\c_{1}=-\frac{1}{2} \\\\v_{1}=x\,axis\,value=-2 \\\\c_{2}=1 \\\\v_{2}=y\,axis\,value=1 \\\\c_{3}=-\frac{1}{2} \\\\v_{3}=z\,axis\,value=4 \\\\b=0 \\\\\\\textup{gives:} \\\\\left(-\frac{1}{2}\cdot-2\right)+\left(1\cdot1\right)+\left(-\frac{1}{2}\cdot4\right)+0=0 \\\\\\\textup{yielding the balanced equation:} \\\\1+1-2+0=0$

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