# Linear Algebra in the Euclidean Plane

## That course gives you many important skills in linear algebra in dimension 2, the fundamental scope to be ready for linear algebra in any dimension.

The course will provide a comprehensive introduction to the highly beneficial subject of linear algebra. The initial stage involves a detailed exploration of linear algebra in the Euclidean plane. Upon completion of the course, you will have the capability to execute the subsequent mathematical operations.

• Solve systems of two linear equations with two variables.
• Utilise, graph, and manipulate vectors in the plane.
• Compute and apply dot products and norms of vectors in the Euclidean plane.
• Define and manipulate angles, along with their trigonometric functions.
• Determine and utilize polar coordinates of a nonzero vector.
• Work with and manipulate complex numbers, visualizing their vector representations.
• Perform calculations involving matrices of linear transformations.
• Identify bases in the plane and compute transition matrices for changing bases.
• Compute eigenvalues and eigenvectors of a matrix, utilising them for matrix diagonalisation.
• Lastly, diagonalise matrices in ℂ that have no real eigenvalues.

All of these concepts will be taught from a practical perspective, incorporating graphics and Python sessions and scripts.

Fabienne Chaplais

Fabienne holds the Agregation in Mathematics, the highest French diploma to teach mathematics in secondary schools. After a 26-year career as an R&D engineer in applied mathematics, she founded Mathedu (Mathematics Re-engineering) with her husband and collaborator Francois, to teach mathematics from a practical point of view, based on Python programming. She is both an expert in mathematics education and a passionate woman who will help you experience the joy of improving your mastery of mathematics.

## Course Curriculum

Preliminaries
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Example of System of 2 Equations of 2 Variables
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The Systems of 2 Linear Equations in 2 Variables
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The Vectors in the Euclidean Plane
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The Dot Product and the Norm of Vectors
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Angles in Radian and Trigonometric Functions
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The Polar Coordinates and the Rotations
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The Complex Numbers as Vectors
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The Linear Mappings and their Matrices
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Basis and Change of Basis
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Symmetric, Skew Symmetric and Unit Matrices
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Eigenvalues and Eigenvectors
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Diagonalisation in the Field of Complex Numbers
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Final Assessment
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Conclusion
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