## Linear AlgebraCovers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers. |

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### Contents

DETERMINANTS | 1 |

LINEAR SPACES | 31 |

SYSTEMS OF LINEAR EQUATIONS | 58 |

LINEAR FUNCTIONS OF A VECTOR ARGUMENT | 75 |

COORDINATE TRANSFORMATIONS | 118 |

THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR | 133 |

BILINEAR AND QUADRATIC FORMS | 179 |

EUCLIDEAN SPACES | 214 |

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### Common terms and phrases

annihilating polynomial arbitrary axioms basis elt basis minor bilinear form A(x blocks called canonical basis canonical coefficients canonical form category JT coincides complex numbers components conjugate consisting construct coordinate corresponding defined denote diagonal dimension direct sum eigenvalues eigenvectors elements epimorphism equal Euclidean space example fact finite-dimensional formula functions given hence Hermitian Hint hyperplane invariant subspaces inverse isomorphic Lemma linear combination linear manifold spanned linear operator linear space linearly independent minors of order Moreover morphism multiplication n-dimensional space nonsingular nonzero obtain obviously operator mapping orthogonal orthonormal basis positive definite problem Proof proved quadratic form A(x rank real numbers respect scalar product semisimple algebra simple algebra solution space K space Kn Suppose surface symmetric bilinear form tensor Theorem transformation two-sided ideal unitary space values vanish vector x e vectors xlt xt whole space