## A Short Course in Matrix TheoryPresents the essentials of matrix theory and linear algebra as quickly and painlessly as possible, with the intent of reaching the topics of eigenvalues, eigenvectors, diagonalization, and some applications within the time constraints of a short course. Concepts and techniques are introduced by exam |

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### Contents

1 | |

3 | |

7 | |

Section 13 Reduced Row Echelon Form | 12 |

Section 14 Inverse | 21 |

Section 15 Rank and Dimension | 27 |

Section 16 Null Space and Hermite Form | 33 |

Determinants | 39 |

Applications | 83 |

Section 42 Differential Equations | 91 |

Section 43 Powers of a Matrix and the Fibonacci Sequence | 96 |

Section 44 Least Squares Approximation | 99 |

Geometry | 103 |

Section 52 Lines in R | 107 |

Section 53 Projection | 113 |

Section 54 Orthogonal Matrices and Quadrics | 115 |

### Common terms and phrases

augmented matrix basis of Rn basis vectors called characteristic polynomial column space column vectors coordinates determinant diagonal matrix diagonalizable dimension dot product eigenspace eigenvectors elementary row operations Enter key entry Example Find Example Let Exercises Fact find a basis find an orthogonal Find the equation find the RREF following matrices form a basis formula geometric multiplicity gives Hermite form hit the Enter identity matrix invertible matrix linear combination linear transformation linearly independent set matrix equation matrix named n x n matrix null space orthogonal basis orthogonal matrix orthogonally diagonalizes orthonormal basis parametric equations plane preserves length Rank(A real eigenvalues reduced row echelon rotation row echelon form row space Section set of vectors solution of Ax Solve the system span R2 spanning set square matrix standard basis subspace of Rn system of equations u x v unique upper triangular vectors in Rn xi(t zero row zero vector