Linear AlgebraThree-dimensional analytic geometry; Vector spaces; Matrices and systemsj of linear equations; Determinants; Inner products; Linear transformations; Similarity and diagnalization theorems; Reduction of quadratic forms and applications; Complex inner product spaces. |
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a₁ algebra arbitrary augmented matrix b₁ basis F1 c₁ C₂ called coefficients column vectors complex numbers coordinate system coordinate vector Corollary defined denoted diagonal matrix dimension E₁ eigenvalues eigenvectors Example Exercise F-basis F₁ F₂ Find finite dimensional follows Hence Hermitian Hermitian matrix inner product space intersection inverse isomorphism ith row jth column k₁ k₂ L₁ linear transformation linearly independent mapping mutually orthogonal n-dimensional vector space nonsingular matrix nonzero vector null space one-to-one orthogonal matrix orthonormal basis P-¹AP permutation plane polynomials positive definite Proof Prove quadratic form quadric R₁ R₂ rank real numbers reduced row-echelon relative rotation S₁ S₂ satisfy scalar multiple Section set of vectors solution space solution vector space over F space spanned standard basis subspace surface symmetric Theorem transition matrix unique unit vectors unitary unitary space V₁ V₂ vectors X1 X₁ XTAX xy-plane Y₁ zero vector