Add Linear Algebra Documentation - Initial Vectors Section (#10)

* Add comprehensive Linear Algebra documentation

Create LinearAlgebra documentation section with four foundational topics:
- 01 Vectors.md: Operations, magnitude, dot product, orthogonality, span
- 02 Lines.md: Parametric, implicit, explicit forms; conversions; intersections
- 03 Linear Transformations.md: Matrix representation, rotation, scaling, reflection, projection
- 04 Matrices.md: Operations, determinant, inverse, eigenvalues, decompositions

All files follow established documentation patterns:
- MathWorld links for mathematical terms
- Proper LaTeX formatting for equations
- Cross-references between related topics
- Applications sections

* Rename directory to "Basics"

* Add Systems of Equations docs and expand Matrices content

- Add 05 Systems of Equations.md covering algebraic methods (substitution, elimination), nonlinear systems, and solution types
- Expand 04 Matrices.md with Cramer's rule example, matrix inverse method example, and detailed examples for infinitely many solutions and homogeneous systems
- Update make.jl to include new Systems of Equations page
- Minor updates to 01 Vectors.md

* Expand Vectors documentation with notation and components sections

- Reorder definitions (algebraic first, then geometric)
- Add Vector Notation section (column/row vectors, boldface, hat notation)
- Add Components and Dimensions section
- Fix typo in R^2 definition

* Rename notebook

* Add starter basics notebook

* add markdown lint ignore file

* add modified copilot instructions

* Add modified file name

* Fix MathJax3 display math rendering and update documentation patterns

- Add \[...\] delimiters to MathJax3 displayMath config (fixes literal \[ \] appearing)
- Fix bullet points where LaTeX started immediately after dash (Unit Vectors, Polar Form, Key insights)
- Update copilot-instructions.md with:
  - Enhanced bullet point rule with more examples
  - Absolute value notation (\lvert/\rvert)
  - Complete MathJax3 configuration section with migration instructions
  - Mark as canonical basis for other Julia Math repos

* Add vector operations examples and generalize to R^n

Vectors documentation:
- Add section on generalizing vector operations to R^n
- Include addition, scalar multiplication, subtraction formulas for n-dimensional vectors
- Add notes on applications in data science and machine learning

Basics notebook:
- Add vector operation examples with production quantities
- Add complex vector expression examples (v₁ - 4v₂ + 2v₃)

* Add Basis Vectors section with simplified explanation

- Define basis vectors without using 'vector space' or 'linearly independent'
- Show standard basis for R^2 and R^3 with concrete examples
- Note connection between basis vectors and unit vectors
- Fix L2 norm formatting

* Add 'define before use' principle and refine Vectors doc

- Add documentation guideline: all terms must be defined before use
- This principle drives document structure and section ordering
- Refine 'Components and Dimensions' section to define space/R^n
- Ensure logical progression throughout Vectors documentation

* move dot product section back after definition of magnitide and direction

* Add derivations and explanations for vector formulas

- Add Pythagorean derivation for magnitude formula
- Explain extension to R^n via recursive Pythagorean theorem
- Add trigonometric derivation for polar form
- Explain normalization formula (why dividing by magnitude works)
- Extend basis vectors explanation to R^n with summation notation

* Reorganize Vectors doc for logical flow and add notebook example

- Rearrange sections to follow 'define before use' principle
- Add example to Basics notebook

* Add comprehensive Geometry documentation section

- Add 10 new Geometry documents covering 2D/3D linear algebra:
  * Points and Coordinates (vectors, basis, barycentric coords)
  * Lines (parametric, implicit, explicit forms)
  * Dot Product and Orthogonality
  * Projections and Distance
  * Line Intersections
  * Transformations (rotation, reflection, projection, scaling)
  * Matrices and Maps (matrix-vector multiplication, determinants)
  * Linear Systems (Cramer's rule, Gaussian elimination)
  * Inverse and Kernel (invertibility, null space)
  * 3D Geometry (cross product, planes, 3D transformations)

- Add 88 supporting images in Geometry/assets/
- Update make.jl with Geometry section in navigation
- Add pagesonly=true to exclude source reference files
- Update .gitignore to exclude Linear Algebra Capacities folder

* Coordinate Basics and Geometry 01 documents

Basics/01 Vectors:
- Remove geometric concepts (magnitude, distance, basis vectors)
- Keep algebraic definitions and properties
- Add cross-references to Geometry section

Geometry/01 Points and Coordinates:
- Add position vector, free vector, displacement vector definitions
- Add parallel vectors section
- Add cross-reference to Basics for vector operations
- Fix image paths

* Coordinate Basics/01 and Geometry/01: separate algebraic vs geometric content

- Remove geometric interpretations from Basics/01 Vectors.md (parallelogram rule, scaling, linear independence)
- Add Linear Independence (Geometric View) section to Geometry/01 Points and Coordinates.md
- Update all cross-references to include specific section anchors
- Add bidirectional links between algebraic and geometric sections

* Fix cross-document section links with proper anchor syntax

- Use correct capitalization in section anchors (e.g., #Coordinate-Independent-Operations)
- Add document context to cross-references (e.g., 'in the Points and Coordinates document')
- Document cross-reference syntax in copilot-instructions.md for future reference
- Spaces in file paths use actual spaces, spaces in anchors use hyphens

* emove Basics/02 Lines.md and renumber remaining Basics docs

- Delete Basics/02 Lines.md (content covered in Geometry section)
- Add skew lines section to Geometry/10 3D Geometry.md
- Renumber Basics files: 03→02, 04→03, 05→04
- Update make.jl pages array with new file names
- Fix all cross-references to use updated file numbers

* Prepare initial PR with Vectors documentation only

- Update index.md with source references (Farin & Hansford text)
- Comment out all pages except 01 Vectors in make.jl for initial deploy
- Comment out cross-references in 01 Vectors.md to excluded pages
- Enables incremental review: deploy Vectors first, add more as reviewed

* Fix Julia 1.12 compatibility and update index.md

Symbolics.jl fix:
- Add fold=Val{true}() to substitute calls to force evaluation
- Add Float64.() wrapper to ensure numeric matrix types
- Fixes rtoldefault MethodError in isapprox comparisons
- All 68 tests now pass on Julia 1.12

index.md updates:
- Add detailed overview of package purpose and source materials
- Document source books for Basics and Geometry sections
- Minor grammar and clarity improvements

Author: AronT-TLV

.github/copilot-instructions.md

@@ -1,8 +1,10 @@
 # Copilot Instructions for Linear_Algebra
 
-## Project Architecture
+> **Note:** These instructions serve as the canonical basis for other repositories in the Julia Math study project. Documentation patterns, LaTeX conventions, and MathJax3 configuration documented here should be referenced by related repositories.
 
-This is a **Julia linear algebra computing project** using DrWatson for reproducibility, focused on geometric transformations and linear algebra operations with visualization capabilities. The codebase follows a **modular mathematical library pattern** with comprehensive testing and documentation.
+## Project Overview
+
+This is a **Julia linear algebra project** using DrWatson for reproducibility, focused on geometric transformations and linear algebra operations with visualization capabilities. The codebase follows a **modular mathematical library pattern** with comprehensive testing and documentation.
 
 ### Core Components
 
@@ -14,7 +16,10 @@ This is a **Julia linear algebra computing project** using DrWatson for reproduc
 
 ## Critical Development Patterns
 
-### Module Structure
+### Module Structure & Exports
+
+All code uses `@reexport` pattern and exports both computational + plotting functions:
+
 ```julia
 # Main module uses @reexport for clean interface
 using Reexport
@@ -38,12 +43,17 @@ export distance_2_points, center_of_gravity, barycentric_coord, plot_param_line
 # ... matrix functions, line functions, etc.
 ```
 
-### Environment-Aware Module Loading
+### Automatic CI/Interactive Detection
+
+Module auto-configures at load time - no manual intervention needed:
+
 ```julia
-# The module automatically detects CI vs interactive environments
+# src/Math_Foundations.jl - Runs on module load
 if haskey(ENV, "CI") || get(ENV, "GKSwstype", "") == "100"
     ENV["GKSwstype"] = "100"  # Headless plotting
     gr(show=false)
+else
+    gr()  # Interactive plotting
 end
 ```
 
@@ -312,14 +322,131 @@ distance_to_implicit_line(a::Number, b::Number, c::Number, r::Point) -> Float64
 foot_of_line(P::Point, v::Vector, R::Point) -> Tuple(Point, Float64)
 ```
 ## Documentation Patterns
-- Use LaTex for all mathematical notation
-- Use Markdown for explanations
-- After creating or editing a markdown document always review and fix all linting issues, unless the document is a configuration file of some kind
+
+### Structure for Mathematical Concept Documentation
+
+Documentation in `docs/src/` explains general math concepts (not code). Follow these patterns:
+
+**Document Organization:**
+- Start with concept overview and [MathWorld](https://mathworld.wolfram.com/) link in opening paragraph
+- Use hierarchical headings: `##` for major topics, `###` for subtopics
+- Group related concepts logically
+- Include real-world applications section
+
+**Content Style:**
+- **Definitions first**: Clear, precise mathematical definitions with MathWorld links
+- **Define before use**: All terms must be defined before they are used elsewhere in the document. This principle drives the order and structure of sections—ensure logical progression where each concept builds only on previously defined terms.
+- **Build progressively**: Simple concepts → complex relationships → applications
+- **Multiple representations**: Equations, tables, visual aids (SVG diagrams when helpful)
+- **Context matters**: Explain _why_ concepts are important, not just _what_ they are
+- **Derivations**: Show mathematical reasoning step-by-step
+
+**Mathematical Notation:**
+- Use LaTeX: `$$` for display equations, inline with `$...$`
+- **CRITICAL**: `$$` display blocks in Documenter.jl must have `$$` directly attached to content (no blank lines or return after opening or before closing)
+  - ✅ **CORRECT** (multi-line aligned block):
+    ```
+    $$\begin{aligned}
+    b &= a\sqrt{1-e^2} \\
+    c &= ae \\
+    a^2 &= b^2 + c^2
+    \end{aligned}$$
+    ```
+  - ✅ **CORRECT** (single equation): `$$equation$$`
+  - ❌ **WRONG** (blank lines or return between `$$` and content):
+    ```
+    $$
+
+    content
+
+    $$
+    ```
+- **Bullet point style**: List items starting with `-` must begin with **text**, then LaTeX math. LaTeX commands immediately after `-` render incorrectly (misaligned or broken formatting)
+  - ✅ **CORRECT**: `- The semi-major axis is $a$`
+  - ✅ **CORRECT**: `- stretch the vector: $\lvert c \rvert \gt 1$`
+  - ❌ **WRONG**: `- $a$ is the semi-major axis` (LaTeX immediately after dash)
+  - ❌ **WRONG**: `- $\mathbf{e}_1 = \lbrack 1, 0 \rbrack$ (points along x-axis)` (LaTeX starts the line)
+- **Equation placement style**: Prefer inline equations with "where:" at end of sentence, followed by list
+  - ✅ **CORRECT**: `For an ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where:`
+  - ❌ **AVOID**: Blank line, then equation block, then blank line, then "where:"
+- LaTeX syntax for symbols: `^\circ` not `°`, `\frac{}{}` for fractions
+- Label variables clearly: "where: $r$ = radius, $θ$ = angle"
+- Use aligned equations: `\begin{aligned}...\end{aligned}` for multi-step derivations
+- **Square brackets in LaTeX math**: Use `\lbrack` and `\rbrack` instead of `[` and `]` inside math expressions to avoid markdown link interpretation errors
+  - ✅ **CORRECT**: `$\mathbf{v} = \lbrack v_1, v_2 \rbrack$`
+  - ❌ **WRONG**: `$\mathbf{v} = [v_1, v_2]$` (markdown interprets `[v_1, v_2]` as a link)
+- **Absolute value notation**: Use `\lvert` and `\rvert` instead of `|` to avoid markdown pipe interpretation
+  - ✅ **CORRECT**: `$\lvert c \rvert > 1$`
+  - ❌ **WRONG**: `$|c| > 1$` (pipe may be interpreted as table delimiter)
+
+**MathJax3 Configuration (Linear_Algebra uses MathJax3 instead of default KaTeX):**
+
+This project uses MathJax3 for math rendering. Key configuration in `docs/make.jl`:
+
+```julia
+mathengine = Documenter.MathJax3(Dict(
+    :loader => Dict("load" => ["[tex]/physics", "[tex]/ams"]),
+    :tex => Dict(
+        "packages" => ["base", "ams", "mathtools"],
+        "inlineMath" => [["\$", "\$"]],
+        "displayMath" => [["\$\$", "\$\$"], ["\\[", "\\]"]],
+    ),
+)),
+```
+
+**Critical**: The `displayMath` array must include **both** `$$...$$` AND `\[...\]` delimiters because:
+- Markdown source uses `$$...$$` for display math
+- Documenter.jl converts some `$$` blocks to `\[...\]` in generated HTML (especially in list items or new paragraphs)
+- Without both delimiters, MathJax won't process `\[...\]` blocks and they render as literal text
+
+**To switch from KaTeX to MathJax3:**
+1. Add `mathengine` parameter to `Documenter.HTML()` in `docs/make.jl`
+2. Include both delimiter styles in `displayMath`
+3. Load desired packages (ams, mathtools, physics) via `:loader` and `"packages"`
+4. Clean rebuild: `rm -rf docs/build && julia --project=. docs/make.jl`
+
+**MathWorld Links:**
+- Link every new mathematical term on first mention
+- Format: `[Term](https://mathworld.wolfram.com/Term.html)`
+- Verify URLs before adding (use fetch_webpage)
+- Compare multiple URLs to choose most appropriate
+
+**Visual Elements:**
+- SVG diagrams for geometric concepts
+- Tables for reference data (common values, conversions)
+- Consistent styling in diagrams (colors, labels, annotations)
+
+**Markdown Conventions:**
+- _Underscore_ for emphasis (not asterisk)
+- **Bold** for important terms and section labels
+- Fix all linting issues after editing (except config files)
+- Code blocks with language tags: ````julia` for Julia examples
 - Follow the pattern of existing function documentation in src directory
-- When a new mathematical concept is introduced, add an external link to [MathWorld](https://mathworld.wolfram.com/) for that specific concept.
-  - Compare multiple potential URLs to choose the most appropriate one
-  - Avoid replacing working links with inferior alternatives
-  - Always use fetch_webpage function to check the link is valid and points to the correct concept.
+
+**Cross-Document Section Links (Documenter.jl):**
+
+When linking to a specific section in another document, use standard markdown link syntax with the full file path and section anchor:
+
+```markdown
+[Display Text](../Folder/Document Name.md#Section-Heading)
+```
+
+**CRITICAL rules for section anchors:**
+- Section anchors must **exactly match** the heading text with proper capitalization
+- Spaces in **file names** remain as spaces (not `%20`)
+- Spaces in **section anchors** become hyphens (`-`)
+- Parentheses in section names are preserved in the anchor
+
+**Examples:**
+- ✅ **CORRECT**: `[Coordinate Independent Operations](../Geometry/01 Points and Coordinates.md#Coordinate-Independent-Operations)`
+- ✅ **CORRECT**: `[Linear Independence (Geometric View)](../Geometry/01 Points and Coordinates.md#Linear-Independence-(Geometric-View))`
+- ❌ **WRONG**: `[Section](../Geometry/01%20Points%20and%20Coordinates.md#section)` (wrong: `%20` and lowercase)
+- ❌ **WRONG**: `[Section](@ref "Section Name")` (`@ref` is for docstrings, not cross-document links)
+
+**Best practice for cross-references:**
+- Include context about which document contains the section: "see [Section Name](path#anchor) in Document Name"
+- Verify the section heading spelling and capitalization before creating the link
+- Test the link by building docs (`julia --project=. docs/make.jl`) and clicking in browser
 
 ## Communication Patterns
 

.gitignore

@@ -417,4 +417,7 @@ $RECYCLE.BIN/
 ## Acknowledgements
 # Many thanks to `https://gitignore.io/`, written and maintained by Joe Blau, which contributed much material to this gitignore file.
 
-/Manifest.toml
+/Manifest.toml
+docs/src/Linear\ Algebra\ Old/
+docs/src/Linear Algebra Capacities/
+

.markdownlint-cli2.jsonc

@@ -0,0 +1,5 @@
+{
+    "ignores": [
+        ".github/copilot-instructions.md"
+    ]
+}

docs/make.jl

@@ -6,15 +6,45 @@ using Linear_Algebra
 makedocs(;
     modules=[Linear_Algebra],  # Specify your module here
     sitename = "Linear_Algebra",
+    # pagesonly=true excludes all markdown files not listed in pages (e.g., Linear Algebra Capacities/)
+    pagesonly = true,
     # This argument is only so that the sequence of pages in the sidebar is configured
-    # By default all markdown files in `docs/src` are expanded and included.
-    pages = ["index.md", "sites.md"],
+    pages = [
+        "index.md",
+        "Basics" => [
+            "Basics/01 Vectors.md",
+            # "Basics/02 Linear Transformations.md",
+            # "Basics/03 Matrices.md",
+            # "Basics/04 Systems of Equations.md",
+        ],
+        "Geometry" => [
+            "Geometry/01 Points and Coordinates.md",
+            # "Geometry/02 Lines.md",
+            # "Geometry/03 Dot Product and Orthogonality.md",
+            # "Geometry/04 Projections and Distance.md",
+            # "Geometry/05 Line Intersections.md",
+            # "Geometry/06 Transformations.md",
+            # "Geometry/07 Matrices and Maps.md",
+            # "Geometry/08 Linear Systems.md",
+            # "Geometry/09 Inverse and Kernel.md",
+            # "Geometry/10 3D Geometry.md",
+        ],
+        "sites.md"
+    ],
     # Don't worry about what `CI` does in this line.
     format = Documenter.HTML(
         prettyurls = CI,
         edit_link = "main",
         assets = ["assets/custom.css"],
         inventory_version = "0.1.0",
+        mathengine = Documenter.MathJax3(Dict(
+            :loader => Dict("load" => ["[tex]/physics", "[tex]/ams"]),
+            :tex => Dict(
+                "packages" => ["base", "ams", "mathtools"],
+                "inlineMath" => [["\$", "\$"]],
+                "displayMath" => [["\$\$", "\$\$"], ["\\[", "\\]"]],
+            ),
+        )),
         footer = "Powered by [Documenter.jl](https://documenter.jl) and the [Julia Programming Language](https://julialang.org) generated on $(Dates.format(Dates.now(), "yyyy-mm-dd HH:MM:ss"))"
     ),
 )