 # A First Course in Linear Algebra A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Along the way, determinants and eigenvalues get fair time.
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• Systems of Linear Equations
• What is Linear Algebra?
• Solving Systems of Linear Equations
• Reduced Row-Echelon Form
• Types of Solution Sets
• Homogeneous Systems of Equations
• Nonsingular Matrices
• Vectors
• Vector Operations
• Linear Combinations
• Spanning Sets
• Linear Independence
• Linear Dependence and Spans
• Orthogonality
• Matrices
• Matrix Operations
• Matrix Multiplication
• Matrix Inverses and Systems of Linear Equations
• Matrix Inverses and Nonsingular Matrices
• Column and Row Spaces
• Four Subsets
• Vector Spaces
• Vector Spaces
• Subspaces
• Linear Independence and Spanning Sets
• Bases
• Dimension
• Properties of Dimension
• Determinants
• Determinant of a Matrix
• Properties of Determinants of Matrices
• Eigenvalues
• Eigenvalues and Eigenvectors
• Properties of Eigenvalues and Eigenvectors
• Similarity and Diagonalization
• Linear Transformations
• Linear Transformations
• Injective Linear Transformations
• Surjective Linear Transformations
• Invertible Linear Transformations
• Representations
• Vector Representations
• Matrix Representations
• Change of Basis
• Orthonormal Diagonalization
Supplements: None