diff --git a/.clang-format b/.clang-format
index e993027..38e37ec 100644
--- a/.clang-format
+++ b/.clang-format
@@ -7,6 +7,8 @@ ConstructorInitializerAllOnOneLineOrOnePerLine: true
 PointerAlignment: Left
 SortIncludes: true
 SpacesBeforeTrailingComments: 2
+SeparateDefinitionBlocks: Always
+SpaceBeforeCpp11BracedList: true
 UseTab: Always
 MaxEmptyLinesToKeep: 5
 
diff --git a/.gitea/workflows/ubuntu.yaml b/.gitea/workflows/ubuntu.yaml
new file mode 100644
index 0000000..4f37a72
--- /dev/null
+++ b/.gitea/workflows/ubuntu.yaml
@@ -0,0 +1,29 @@
+name: Linux arm64
+run-name: Build And Test
+
+on: [push]
+
+jobs:
+  Build:
+    runs-on: ubuntu-latest
+    steps:     
+      - name: Check out repository code
+        uses: actions/checkout@v3
+
+      - name: Prepare XMake
+        uses: xmake-io/github-action-setup-xmake@v1
+        with:
+          xmake-version: latest
+          actions-cache-folder: '.xmake-cache'
+          actions-cache-key: 'ubuntu'
+
+      - name: XMake config
+        run: xmake f -p linux -y --root
+
+      - name: Build
+        run: xmake --root
+
+      - name: Test
+        run: |
+          xmake f -m debug --root
+          xmake test --root
diff --git a/.gitignore b/.gitignore
index 1521057..f928f14 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,4 +5,5 @@ build/
 # MacOS Cache
 .DS_Store
 
-
+# VsCode
+.vscode
diff --git a/README.md b/README.md
index 1d1e02d..aa2b67b 100644
--- a/README.md
+++ b/README.md
@@ -2,4 +2,16 @@
 
 
 # Cahier des charges
-![imagecdc](PeiP2_MAM-INFO_projet_02.jpg)
\ No newline at end of file
+![imagecdc](PeiP2_MAM-INFO_projet_02.jpg)
+
+# Build
+
+```
+xmake
+```
+
+# Run
+
+```
+xmake run
+```
\ No newline at end of file
diff --git a/matricies/test/matrice1.test b/matricies/test/matrice1.test
new file mode 100644
index 0000000..9d63899
--- /dev/null
+++ b/matricies/test/matrice1.test
@@ -0,0 +1,12 @@
+2 3
+1 2 3
+4 5 6
+
+2 2
+0 1
+1 0
+
+3 1
+1
+-2
+1
\ No newline at end of file
diff --git a/src/Gauss.cpp b/src/Gauss.cpp
new file mode 100644
index 0000000..60513ee
--- /dev/null
+++ b/src/Gauss.cpp
@@ -0,0 +1,75 @@
+#include "Gauss.h"
+
+#include "Matrix.h"
+
+namespace Gauss {
+
+static void GaussNonJordan(Matrix& mat, bool reduite) {
+	int r = -1;
+	for (std::size_t j = 0; j < mat.GetColumnCount(); j++) {
+		std::size_t indice_ligne_maximum = r + 1;
+
+		// Recherche maximum
+		for (std::size_t i = r + 1; i < mat.GetRawCount(); i++) {
+			if (std::abs(mat.at(i, j)) > std::abs(mat.at(indice_ligne_maximum, j)))
+				indice_ligne_maximum = i;
+		}
+
+		// Si A[k,j]≠0 alors    (A[k,j] désigne la valeur de la ligne k et de la colonne j)
+		if (!IsEqualZero(mat.at(indice_ligne_maximum, j))) {
+			r++;
+
+			// Si k≠r alors
+			if (indice_ligne_maximum != r) {
+				// Échanger les lignes k et r  (On place la ligne du pivot en position r)
+				for (std::size_t k = 0; k < mat.GetColumnCount(); k++) {
+					std::swap(mat.at(indice_ligne_maximum, k), mat.at(r, k));
+				}
+			}
+
+			// Pour i de 1 jusqu'à n               (On simplifie les autres lignes)
+			for (std::size_t i = (reduite ? 0 : j); i < mat.GetRawCount(); i++) {
+				// Si i≠r alors
+				if (i != r) {
+					// Soustraire à la ligne i la ligne r multipliée par A[i,j] (de façon à
+					// annuler A[i,j])
+					for (int k = mat.GetColumnCount() - 1; k >= 0; k--) {
+						long double pivot = mat.at(r, j);
+						long double anul = mat.at(i, j);
+						mat.at(i, k) = mat.at(i, k) * pivot - mat.at(r, k) * anul;
+					}
+				}
+			}
+		}
+	}
+}
+
+static void GaussJordan(Matrix& mat, bool reduite) {
+	GaussNonJordan(mat, reduite);
+	for (std::size_t i = 0; i < mat.GetRawCount(); i++) {
+		int k = -1;
+		for (std::size_t j = 0; j < mat.GetColumnCount(); j++) {
+			if (!IsEqualZero(mat.at(i, j))) {
+				k = j;
+				break;
+			}
+		}
+		// ligne de 0
+		if (k == -1)
+			break;
+		// on divise la ligne par (i, k)
+		long double annul = mat.at(i, k);
+		for (int j = 0; j < mat.GetColumnCount(); j++) {
+			mat.at(i, j) /= annul;
+		}
+	}
+}
+
+void GaussJordan(Matrix& mat, bool reduite, bool normalise) {
+	if (normalise)
+		GaussJordan(mat, reduite);
+	else
+		GaussNonJordan(mat, reduite);
+}
+
+}  // namespace Gauss
\ No newline at end of file
diff --git a/src/Gauss.h b/src/Gauss.h
new file mode 100644
index 0000000..168919a
--- /dev/null
+++ b/src/Gauss.h
@@ -0,0 +1,9 @@
+#pragma once
+
+class Matrix;
+
+namespace Gauss {
+
+void GaussJordan(Matrix& mat, bool reduite, bool normalise);
+
+}  // namespace Gauss
\ No newline at end of file
diff --git a/src/Matrix.cpp b/src/Matrix.cpp
index 27341e2..596a55f 100644
--- a/src/Matrix.cpp
+++ b/src/Matrix.cpp
@@ -1,35 +1,39 @@
+#include "Matrix.h"
+
 #include <algorithm>
+#include <cassert>
 #include <cmath>
 #include <fstream>
 #include <iostream>
 
-#include "Matrix.h"
-
 Matrix::Matrix(const std::string& fileNameInput) {
 	Load(fileNameInput);
 }
 
-Matrix::Matrix(std::size_t lignes, std::size_t colonnes) : m_Lignes(lignes), m_Colonnes(colonnes), m_Dimension(lignes * colonnes) {
-	m_Data.resize(m_Dimension);
+Matrix::Matrix(std::size_t lignes, std::size_t colonnes) : m_Raws(lignes), m_Columns(colonnes) {
+	m_Data.resize(m_Raws * m_Columns);
 }
+
 Matrix::Matrix(std::size_t lignes, std::size_t colonnes, std::initializer_list<long double>&& initList) :
-	m_Lignes(lignes), m_Colonnes(colonnes), m_Dimension(lignes * colonnes) {
+	m_Raws(lignes), m_Columns(colonnes) {
 	m_Data = initList;
-	m_Data.resize(m_Dimension);
+	m_Data.resize(m_Raws * m_Columns);
 }
+
 Matrix::~Matrix() {}
 
 Matrix Matrix::operator*(const Matrix& other) const {
-	if (m_Colonnes != other.m_Lignes) {
+	if (m_Columns != other.m_Raws) {
 		std::cerr << "Mutiplication impossible car la dimensions des matrices est incompatible" << std::endl;
+		return {1, 1, {0}};
 	}
 
-	Matrix result(m_Lignes, other.m_Colonnes);
+	Matrix result(m_Raws, other.m_Columns);
 
-	for (std::size_t i = 0; i < m_Lignes; ++i) {
-		for (std::size_t j = 0; j < other.m_Colonnes; ++j) {
+	for (std::size_t i = 0; i < m_Raws; ++i) {
+		for (std::size_t j = 0; j < other.m_Columns; ++j) {
 			long double sum = 0;
-			for (std::size_t k = 0; k < m_Colonnes; k++) {
+			for (std::size_t k = 0; k < m_Columns; k++) {
 				sum += at(i, k) * other.at(k, j);
 			}
 			result.at(i, j) = sum;
@@ -38,15 +42,11 @@ Matrix Matrix::operator*(const Matrix& other) const {
 	return result;
 }
 
-static bool IsEqualZero(long double var) {
-	return std::abs(var) < std::pow(10, -5);
-}
-
 void Matrix::Print() const {
-	for (size_t i = 0; i < m_Lignes; ++i) {
+	for (size_t i = 0; i < m_Raws; ++i) {
 		std::cout << "[ ";
-		for (size_t j = 0; j < m_Colonnes; ++j) {
-			std::size_t indice = i * m_Lignes + j;
+		for (size_t j = 0; j < m_Columns; ++j) {
+			std::size_t indice = i * m_Raws + j;
 			std::cout << at(i, j) << " ";
 		}
 		std::cout << "]";
@@ -54,16 +54,9 @@ void Matrix::Print() const {
 	}
 }
 
-void Matrix::PrintDebug() {
-#ifndef NDEBUG
-	Print();
-	std::cout << "\n";
-#endif
-}
-
 void Matrix::Insert() {
-	for (size_t i = 0; i < m_Lignes; ++i) {
-		for (size_t j = 0; j < m_Colonnes; ++j) {
+	for (size_t i = 0; i < m_Raws; ++i) {
+		for (size_t j = 0; j < m_Columns; ++j) {
 			std::cin >> at(i, j);
 		}
 		std::cout << std::endl;
@@ -71,142 +64,127 @@ void Matrix::Insert() {
 }
 
 void Matrix::Save(const std::string& fileName) {
-	std::ofstream out{fileName};
+	std::ofstream out {fileName};
 	if (!out) {
 		std::cerr << "Impossible de sauvegarder la matrice !\n";
 		return;
 	}
-	out << m_Lignes << " " << m_Colonnes << "\n";
-	for (std::size_t i = 0; i < m_Lignes; i++) {
-		for (std::size_t j = 0; j < m_Colonnes; j++) {
-			out << at(i, j) << " ";
-		}
-		out << "\n";
-	}
+	out << *this;
 }
 
 void Matrix::Load(const std::string& filename) {
-	std::ifstream in{filename};
+	std::ifstream in {filename};
 	if (!in) {
 		std::cerr << "Impossible de charger la matrice !\n";
 		return;
 	}
-	in >> m_Lignes >> m_Colonnes;
-	m_Data.resize(m_Lignes * m_Colonnes);
-	for (std::size_t i = 0; i < m_Lignes; i++) {
-		for (std::size_t j = 0; j < m_Colonnes; j++) {
-			in >> at(i, j);
-		}
-	}
+	in >> *this;
 }
 
 void Matrix::Transpose() {
-	for (std::size_t i = 0; i < m_Lignes; i++) {
-		for (std::size_t j = i; j < m_Colonnes; j++) {
-			std::swap(at(i, j), at(j, i));
+	Matrix result {m_Columns, m_Raws};
+	for (std::size_t i = 0; i < m_Raws; i++) {
+		for (std::size_t j = 0; j < m_Columns; j++) {
+			result.at(j, i) = at(i, j);
 		}
 	}
+	*this = result;
 }
 
-void Matrix::Identity() {
-	for (std::size_t i = 0; i < m_Lignes; i++) {
-		for (std::size_t j = i; j < m_Colonnes; j++) {
-			if (i != j) {
-				at(i, j) = 0;
-			} else {
-				at(i, j) = 1;
-			}
+Matrix Matrix::Identity(std::size_t taille) {
+	Matrix id {taille, taille};
+	for (std::size_t i = 0; i < taille; i++) {
+		for (std::size_t j = i; j < taille; j++) {
+			id.at(i, j) = (i == j);
 		}
 	}
+	return id;
 }
 
-bool Matrix::IsInversed() const {
-	for (std::size_t i = 0; i < m_Lignes; ++i) {
-		std::size_t j;
-		for (j = 0; j < m_Colonnes; ++j) {
-			if (!IsEqualZero(at(i, j))) {
-				break;
-			}
-			return false;
+void Matrix::Augment(const Matrix& droite) {
+	assert(droite.m_Raws == m_Raws);
+	Matrix temp {m_Raws, m_Columns + droite.m_Columns};
+
+	for (std::size_t i = 0; i < m_Raws; i++) {
+		for (std::size_t j = 0; j < m_Columns; j++) {
+			temp.at(i, j) = at(i, j);
 		}
 	}
+
+	for (std::size_t i = 0; i < m_Raws; i++) {
+		for (std::size_t j = 0; j < droite.m_Columns; j++) {
+			temp.at(i, j + m_Columns) = droite.at(i, j);
+		}
+	}
+
+	*this = temp;
+}
+
+bool Matrix::operator==(const Matrix& other) const {
+	if (m_Raws != other.m_Raws || m_Columns != other.m_Columns)
+		return false;
+
+	for (std::size_t i = 0; i < m_Raws; i++) {
+		for (std::size_t j = 0; j < m_Columns; j++) {
+			if (!IsEqualZero(at(i, j) - other.at(i, j)))
+				return false;
+		}
+	}
+
 	return true;
 }
 
-void Matrix::GaussNonJordan(bool reduite) {
-	int r = -1;
-	for (std::size_t j = 0; j < m_Colonnes; j++) {
-		std::size_t indice_ligne_maximum = r + 1;
-
-		// Recherche maximum
-		for (std::size_t i = r + 1; i < m_Lignes; i++) {
-			if (std::abs(at(i, j)) > std::abs(at(indice_ligne_maximum, j)))
-				indice_ligne_maximum = i;
-		}
-
-		// std::cout << "l'indice du maximum est : " << indice_ligne_maximum << "\n\n";
-
-		// Si A[k,j]≠0 alors    (A[k,j] désigne la valeur de la ligne k et de la colonne j)
-		if (!IsEqualZero(at(indice_ligne_maximum, j))) {
-			r++;
-
-			// PrintDebug();
-
-			// Si k≠r alors
-			if (indice_ligne_maximum != r) {
-				// Échanger les lignes k et r  (On place la ligne du pivot en position r)
-				// std::cout << "On échange les lignes " << indice_ligne_maximum << " et " << r << '\n';
-				for (std::size_t k = 0; k < m_Colonnes; k++) {
-					std::swap(at(indice_ligne_maximum, k), at(r, k));
-				}
-			}
-
-			// Pour i de 1 jusqu'à n               (On simplifie les autres lignes)
-			for (std::size_t i = (reduite ? 0 : j); i < m_Lignes; i++) {
-				// Si i≠r alors
-				if (i != r) {
-					// Soustraire à la ligne i la ligne r multipliée par A[i,j] (de façon à
-					// annuler A[i,j])
-					for (int k = m_Colonnes - 1; k >= 0; k--) {
-						long double pivot = at(r, j);
-						long double anul = at(i, j);
-						at(i, k) = at(i, k) * pivot - at(r, k) * anul;
-					}
-				}
-			}
-		}
-	}
-}
-
-void Matrix::GaussJordan(bool reduite) {
-	GaussNonJordan(reduite);
-	for (std::size_t i = 0; i < m_Lignes; i++) {
-		int k = -1;
-		for (std::size_t j = 0; j < m_Colonnes; j++) {
-			if (!IsEqualZero(at(i, j))) {
-				k = j;
-				break;
-			}
-		}
-		// ligne de 0
-		if (k == -1)
-			break;
-		// on divise la ligne par (i, k)
-		long double annul = at(i, k);
-		for (int j = 0; j < m_Colonnes; j++) {
-			at(i, j) /= annul;
-		}
-	}
-}
-
 long double& Matrix::operator[](std::size_t indice) {
 	return m_Data[indice];
 }
 
 long double& Matrix::at(std::size_t ligne, std::size_t colonne) {
-	return m_Data[ligne * m_Lignes + colonne];
+	return m_Data[ligne * m_Columns + colonne];
 }
 
 long double Matrix::at(std::size_t ligne, std::size_t colonne) const {
-	return m_Data[ligne * m_Lignes + colonne];
-}
\ No newline at end of file
+	return m_Data[ligne * m_Columns + colonne];
+}
+
+std::size_t Matrix::GetRawCount() const {
+	return m_Raws;
+}
+
+std::size_t Matrix::GetColumnCount() const {
+	return m_Columns;
+}
+
+Matrix Matrix::SubMatrix(std::size_t origine_ligne, std::size_t origine_colonne, std::size_t ligne, std::size_t colonne) const {
+	assert(m_Raws >= ligne && m_Columns >= colonne);
+	Matrix result {ligne, colonne};
+
+	for (std::size_t i = 0; i < ligne; i++) {
+		for (std::size_t j = 0; j < colonne; j++) {
+			result.at(i, j) = at(i + origine_ligne, j + origine_colonne);
+		}
+	}
+
+	return result;
+}
+
+std::ostream& operator<<(std::ostream& stream, const Matrix& mat) {
+	stream << mat.m_Raws << " " << mat.m_Columns << "\n";
+	for (std::size_t i = 0; i < mat.m_Raws; i++) {
+		for (std::size_t j = 0; j < mat.m_Columns; j++) {
+			stream << mat.at(i, j) << " ";
+		}
+		stream << "\n";
+	}
+	return stream;
+}
+
+std::istream& operator>>(std::istream& stream, Matrix& mat) {
+	stream >> mat.m_Raws >> mat.m_Columns;
+	mat.m_Data.resize(mat.m_Raws * mat.m_Columns);
+	for (std::size_t i = 0; i < mat.m_Raws; i++) {
+		for (std::size_t j = 0; j < mat.m_Columns; j++) {
+			stream >> mat.at(i, j);
+		}
+	}
+	return stream;
+}
diff --git a/src/Matrix.h b/src/Matrix.h
index ad80018..2d34ded 100644
--- a/src/Matrix.h
+++ b/src/Matrix.h
@@ -1,48 +1,51 @@
 #pragma once
 
+#include <cmath>
+#include <cstddef>
 #include <string>
 #include <vector>
 
 class Matrix {
   private:
-	std::size_t m_Lignes;
-	std::size_t m_Colonnes;
-	std::size_t m_Dimension;
+	std::size_t m_Raws;
+	std::size_t m_Columns;
 	std::vector<long double> m_Data;
 
   public:
 	Matrix(const std::string& fileNameInput);
-	Matrix(std::size_t lignes, std::size_t colonnes);
-	Matrix(std::size_t lignes, std::size_t colonnes, std::initializer_list<long double>&& initList);
+	Matrix(std::size_t raws, std::size_t columns);
+	Matrix(std::size_t raws, std::size_t columns, std::initializer_list<long double>&& initList);
 	~Matrix();
 
-	Matrix operator*(const Matrix& other) const;
-
-	void GaussNonJordan(bool reduite);
-
-	void GaussJordan(bool reduite);
-
-	void Print() const;
-
-	void PrintDebug();
+	std::size_t GetRawCount() const;
+	std::size_t GetColumnCount() const;
 
 	void Insert();
+	void Print() const;
 
 	void Save(const std::string& fileName);
-
 	void Load(const std::string& filename);
 
 	void Transpose();
 
-	void Identity();
+	static Matrix Identity(std::size_t size);
 
-	bool IsInversed() const;
+	void Augment(const Matrix& right);
 
-	long double& operator[](std::size_t indice);
+	Matrix SubMatrix(std::size_t raw_origin, std::size_t column_origin, std::size_t raw, std::size_t column) const;
 
-	long double& at(std::size_t ligne, std::size_t colonne);
+	bool operator==(const Matrix& other) const;
+	Matrix operator*(const Matrix& other) const;
+	long double& operator[](std::size_t index);
 
-	long double at(std::size_t ligne, std::size_t colonne) const;
+	long double& at(std::size_t raw, std::size_t column);
+	long double at(std::size_t raw, std::size_t column) const;
+
+	friend std::ostream& operator<<(std::ostream& stream, const Matrix& mat);
+	friend std::istream& operator>>(std::istream& stream, Matrix& mat);
 };
 
-static bool IsEqualZero(long double var);
\ No newline at end of file
+template <typename T>
+bool IsEqualZero(T var) {
+	return std::abs(var) < std::pow(10, -5);
+}
\ No newline at end of file
diff --git a/src/NR.h b/src/NR.h
index e5992a6..559ddda 100644
--- a/src/NR.h
+++ b/src/NR.h
@@ -9,6 +9,7 @@ class NR {
 
   public:
 	NR() : m_Numerator(0), m_Denominator(1) {}
+
 	NR(int entier) : m_Numerator(entier), m_Denominator(1) {}
 	NR(int numerator, int denominator); //check if denominator != 0
 	void NRset(int numerator, int denominator); //same
@@ -39,4 +40,4 @@ class NR {
 	static void test();
 };
 
-int PGCD(int x, int y);
\ No newline at end of file
+int PGCD(int x, int y);
diff --git a/src/Solver.cpp b/src/Solver.cpp
new file mode 100644
index 0000000..5f05d14
--- /dev/null
+++ b/src/Solver.cpp
@@ -0,0 +1,45 @@
+#include "Solver.h"
+
+#include "Gauss.h"
+
+Solver::Solver(const Matrix& mat) : m_Matrix(mat) {}
+
+Vect Solver::Image() const {
+	Matrix result = m_Matrix;
+	result.Transpose();
+	Gauss::GaussJordan(result, true, true);
+	result.Transpose();
+	return {result};
+}
+
+// https://en.wikipedia.org/wiki/Kernel_(linear_algebra)#Computation_by_Gaussian_elimination
+Vect Solver::Kernel() const {
+	Matrix result = m_Matrix;
+	result.Transpose();
+	result.Augment(Matrix::Identity(result.GetRawCount()));
+	Gauss::GaussJordan(result, true, true);
+	result.Transpose();
+
+	// nombre de colonnes non nulles
+	std::size_t origine_colonne = Vect(result.SubMatrix(0, 0, m_Matrix.GetRawCount(), m_Matrix.GetColumnCount())).GetCardinal();
+
+	return {result.SubMatrix(m_Matrix.GetRawCount(), origine_colonne, result.GetRawCount() - m_Matrix.GetRawCount(),
+		result.GetColumnCount() - origine_colonne)};
+}
+
+VectAffine Solver::TriangularSystem() const {
+	Matrix mat = m_Matrix;
+	Gauss::GaussJordan(mat, true, true);
+
+	Solver solver {mat.SubMatrix(0, 0, mat.GetRawCount(), mat.GetColumnCount() - 1)};
+
+	Vect noyau = solver.Kernel();
+	Matrix origin = mat.SubMatrix(0, mat.GetColumnCount() - 1, mat.GetRawCount(), 1);
+
+	return {noyau, origin};
+}
+
+std::size_t Solver::Rank() const {
+	Vect image = Image();
+	return image.GetCardinal();
+}
diff --git a/src/Solver.h b/src/Solver.h
new file mode 100644
index 0000000..2d6bbf5
--- /dev/null
+++ b/src/Solver.h
@@ -0,0 +1,20 @@
+#pragma once
+
+#include "Vect.h"
+
+class Solver {
+  private:
+	Matrix m_Matrix;
+
+  public:
+	Solver(const Matrix& mat);
+
+	~Solver() {}
+
+	Vect Image() const;
+	Vect Kernel() const;
+
+	VectAffine TriangularSystem() const;
+
+	std::size_t Rank() const;
+};
\ No newline at end of file
diff --git a/src/Vect.cpp b/src/Vect.cpp
new file mode 100644
index 0000000..e3a2368
--- /dev/null
+++ b/src/Vect.cpp
@@ -0,0 +1,90 @@
+#include "Vect.h"
+
+#include "Gauss.h"
+#include "Solver.h"
+#include <cassert>
+#include <iostream>
+
+Vect::Vect(const Matrix& mat) : m_Data(mat) {
+	Simplify();
+}
+
+void Vect::Simplify() {
+	Matrix mat = m_Data;
+	for (std::size_t j = 0; j < mat.GetColumnCount(); j++) {
+		std::size_t i;
+		for (i = 0; i < mat.GetRawCount(); i++) {
+			if (!IsEqualZero(mat.at(i, j)))
+				break;
+		}
+		if (i == mat.GetRawCount()) {
+			m_Data = mat.SubMatrix(0, 0, mat.GetRawCount(), j);
+			return;
+		}
+	}
+	m_Data = mat;
+}
+
+std::size_t Vect::GetCardinal() const {
+	return m_Data.GetColumnCount();
+}
+
+bool Vect::operator==(const Vect& other) const {
+	if (GetDimension() != other.GetDimension() || GetCardinal() != other.GetCardinal())
+		return false;
+
+	// on vérifie si chaque vecteur de la deuxième base appartient à la première base
+	for (std::size_t i = 0; i < GetCardinal(); i++) {
+		Vect base = *this;
+		base.AddVector(other.m_Data.SubMatrix(0, i, GetDimension(), 1));
+		if (base.GetCardinal() != GetCardinal())
+			return false;
+	}
+	return true;
+}
+
+void Vect::AddVector(const Matrix& mat) {
+	m_Data.Augment(mat);
+	m_Data.Transpose();
+	Gauss::GaussJordan(m_Data, false, false);
+	m_Data.Transpose();
+	Simplify();
+}
+
+bool Vect::operator!=(const Vect& other) const {
+	return !(*this == other);
+}
+
+Matrix Vect::GetLinearSystem() const {
+	Matrix vect = m_Data;
+	vect.Transpose();
+
+	Solver solver {vect};
+	vect = solver.Kernel().m_Data;
+	vect.Transpose();
+	return vect;
+}
+
+void Vect::Print() const {
+	std::cout << "Espace vectoriel de dimension " << GetCardinal() << " de base :\n\n";
+	for (std::size_t i = 0; i < m_Data.GetRawCount(); i++) {
+		for (std::size_t j = 0; j < m_Data.GetColumnCount(); j++) {
+			std::cout << "[ " << m_Data.at(i, j) << " ]\t";
+		}
+		std::cout << "\n";
+	}
+}
+
+std::size_t Vect::GetDimension() const {
+	return m_Data.GetRawCount();
+}
+
+VectAffine::VectAffine(const Vect& base, const Matrix& origine) :
+	m_Base(base), m_Origin(origine.SubMatrix(0, 0, m_Base.GetDimension(), 1)) {}
+
+void VectAffine::Print() const {
+	std::cout << "\tEspace Affine :\n\n";
+	m_Base.Print();
+	std::cout << "\nOrigine :\n\n";
+	m_Origin.Print();
+}
diff --git a/src/Vect.h b/src/Vect.h
new file mode 100644
index 0000000..64b587e
--- /dev/null
+++ b/src/Vect.h
@@ -0,0 +1,58 @@
+#pragma once
+
+#include "Matrix.h"
+
+// espace vectoriel
+class Vect {
+  private:
+	Matrix m_Data;
+
+  public:
+	/**
+	 * \brief Construit une base d'un espace vectoriel à partir des colonnes d'une matrice.
+	 * Ne prend pas en compte les colonnes de 0
+	 * \param mat Une matrice échelonnée.
+	 */
+	Vect(const Matrix& mat);
+
+	/**
+	 * \brief Affiche la base de l'espace vectoriel dans la console
+	 */
+	void Print() const;
+
+	std::size_t GetDimension() const;
+	std::size_t GetCardinal() const;
+
+	Matrix GetLinearSystem() const;
+
+	/**
+	 * \brief Concatène la base actuelle avec un nouveau vecteur
+	 * \param mat Une matrice colonne de taille GetDimension()
+	 */
+	void AddVector(const Matrix& mat);
+
+	bool operator==(const Vect& other) const;
+	bool operator!=(const Vect& other) const;
+
+  private:
+	void Simplify();
+};
+
+class VectAffine {
+  private:
+	Vect m_Base;
+	Matrix m_Origin;
+
+  public:
+	VectAffine(const Vect& base, const Matrix& origin);
+
+	void Print() const;
+
+	const Vect& GetBase() const {
+		return m_Base;
+	}
+
+	const Matrix& GetOrigin() const {
+		return m_Origin;
+	}
+};
\ No newline at end of file
diff --git a/src/main.cpp b/src/main.cpp
index 4b92d2c..c77b9d1 100644
--- a/src/main.cpp
+++ b/src/main.cpp
@@ -1,9 +1,11 @@
 #include "Matrix.h"
 #include "NR.h"
+#include "Gauss.h"
+#include "Solver.h"
 #include <iostream>
 
 void test() {
-	Matrix mat{"matrice4x4.mat"};
+	/* Matrix mat{"matrice4x4.mat"};
 	mat.Print();
 	// mat.Save("matrice3x3.mat");
 	std::cout << "sdfdjiofoseifheoiefhoig\n";
@@ -15,7 +17,27 @@ void test() {
 	mat.Transpose();
 	std::cout << "<<\nTransposée:\n";
 	mat.Print();
-	// mat.Save("matrice4x4echelonne.mat");
+	// mat.Save("matrice4x4echelonne.mat"); */
+
+	Matrix mat2 {"matrice4x4.mat"};
+	mat2.Print();
+
+	Solver solver {mat2};
+
+	Vect image = solver.Image();
+	Vect noyau = solver.Kernel();
+
+	std::cout << "\tImage :\n";
+	image.Print();
+	std::cout << "Système :\n";
+	image.GetLinearSystem().Print();
+	std::cout << "\tNoyau :\n";
+	noyau.Print();
+	std::cout << "Système :\n";
+	noyau.GetLinearSystem().Print();
+
+	std::cout << "\n\n";
+	solver.TriangularSystem().Print();
 }
 
 void prompt() {
@@ -33,7 +55,7 @@ void prompt() {
 
 	mat.Print();
 
-	mat.GaussJordan(true);
+	Gauss::GaussJordan(mat, true, true);
 
 	mat.Print();
 }
diff --git a/test/test_jordan.cpp b/test/test_jordan.cpp
new file mode 100644
index 0000000..d61e51a
--- /dev/null
+++ b/test/test_jordan.cpp
@@ -0,0 +1,47 @@
+#include "Gauss.h"
+#include "Matrix.h"
+#include <cassert>
+
+#ifdef NDEBUG
+#error "Il faut être en debug mode ! xmake f -m debug"
+#endif
+
+struct Test {
+	Matrix mat;
+	Matrix res;
+};
+
+static const std::vector<Test> TEST_MATRICES = {
+    // test 1
+    {{3, 3, {
+        1, 2, 3,
+        4, 5, 6,
+        7, 8, 9,
+    }}, {3, 3, {
+        1, 0, -1,
+        0, 1, 2,
+        0, 0, 0,
+    }}},
+    // test 2
+    {{3, 3, {
+        4, 5, 6,
+        1, 2, 3,
+        7, 8, 9,
+    }}, {3, 3, {
+        1, 0, -1,
+        0, 1, 2,
+        0, 0, 0,
+    }}}
+};
+
+void test() {
+	for (Test test : TEST_MATRICES) {
+		Gauss::GaussJordan(test.mat, true, true);
+		assert(test.mat == test.res);
+	}
+}
+
+int main(int argc, char** argv) {
+	test();
+	return 0;
+}
diff --git a/test/test_solver.cpp b/test/test_solver.cpp
new file mode 100644
index 0000000..cfe6e7e
--- /dev/null
+++ b/test/test_solver.cpp
@@ -0,0 +1,31 @@
+#include <cassert>
+#include <filesystem>
+#include <fstream>
+#include <iostream>
+
+#include "Solver.h"
+
+namespace fs = std::filesystem;
+
+int main() {
+	std::string path = "test";
+	for (const auto& entry : fs::directory_iterator(path)) {
+		std::string fileName = entry.path().string();
+
+		std::cout << "Opening " << fileName << " ...\n";
+
+		std::ifstream in {fileName};
+
+		Matrix mat {1, 1}, imageMat {1, 1}, noyauMat {1, 1};
+		in >> mat >> imageMat >> noyauMat;
+
+		Vect image {imageMat};
+		Vect noyau {noyauMat};
+
+		Solver solver {mat};
+
+		assert(solver.Image() == image);
+		assert(solver.Kernel() == noyau);
+	}
+	return 0;
+}
\ No newline at end of file
diff --git a/test/test_vect.cpp b/test/test_vect.cpp
new file mode 100644
index 0000000..66a261b
--- /dev/null
+++ b/test/test_vect.cpp
@@ -0,0 +1,31 @@
+#include "Vect.h"
+#include <cassert>
+
+int main() {
+	Vect vect1 {{3, 2, {
+		1, 2,
+		3, 4,
+		5, 6,
+	}}};
+	Vect vect2 {{3, 2, {
+		1, 0,
+		0, 0,
+		0, 1,
+	}}};
+	Vect vect3 {{3, 2, {
+		1, 3,
+		3, 7,
+		5, 11,
+	}}};
+	Vect vect4 {{3, 2, {
+		1, 0,
+		0, 0,
+		1, 11,
+	}}};
+	assert(vect1 == vect3);
+	assert(vect2 == vect4);
+	assert(vect1 != vect2);
+	assert(vect2 != vect3);
+	assert(vect3 != vect4);
+	return 0;
+}
\ No newline at end of file
diff --git a/xmake.lua b/xmake.lua
index 769129f..1744eb4 100644
--- a/xmake.lua
+++ b/xmake.lua
@@ -1,10 +1,43 @@
 add_rules("mode.debug", "mode.release")
 
+set_languages("c++17")
+
+-- Solver Library
 target("Pivot")
-    set_kind("binary")
+    set_kind("static")
     add_files("src/*.cpp")
+    remove_files("src/main.cpp")
+
+
+
+
+
+-- Solver Main
+target("PivotMain")
     set_rundir("$(projectdir)/matricies")
-    set_languages("c++17")
+    add_files("src/main.cpp")
+    add_deps("Pivot")
+    set_default(true)
+
+
+
+
+
+-- Solver tests
+for _, file in ipairs(os.files("test/test_*.cpp")) do
+    local name = path.basename(file)
+    target(name)
+        set_kind("binary")
+        add_files("test/" .. name .. ".cpp")
+        set_rundir("$(projectdir)/matricies")
+        add_includedirs("src")
+
+        set_default(false)
+
+        add_deps("Pivot")
+
+        add_tests("compile_and_run")
+end
 
 --
 -- If you want to known more usage about xmake, please see https://xmake.io