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. |
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
a₁ annihilating polynomial arbitrary axioms b₁ b₂ basis e₁ basis minor bilinear form A(x blocks C₁ called canonical basis canonical coefficients canonical form coincide complex numbers components conjugate consisting construct coordinate corresponding defined denote diagonal dimension direct sum eigenvalues eigenvectors elements epimorphism equal Euclidean space example f₁ fact finite-dimensional formula functions given hence Hermitian Hint hyperplane invariant subspaces inverse isomorphic jth column k₁ Lemma linear combination linear manifold spanned linear operator linear space linearly independent M₁ matrix 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 Suppose surface symmetric bilinear form T₁ tensor Theorem transformation two-sided ideal unitary space values vanish vectors X1 whole space x₁ y₁