Matrices and TransformationsThis text stresses the use of matrices in study of transformations of the plane. Familiarizes reader with role of matrices in abstract algebraic systems and illustrates its effective use as mathematical tool in geometry. Includes proofs of most theorems. Answers to odd-numbered exercises. |
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a₁ a₁₁ angle augmented matrix b₁ c₁ called characteristic equation complex numbers Consider corresponding elements diagonal elements diagonal matrix eigenvalues eigenvector associated eigenvectors elementary row transformation elements of row equal Example 1 Determine Exercises In Exercises exists Hamilton-Cayley Theorem Hence Hermitian matrix image points k₁ linear equations linear homogeneous transformation linearly independent maps each point matrix equation matrix F matrix of order matrix representing multiplicative inverse multiply the elements nonsingular matrix nonzero scalar one-to-one mapping order three orthogonal matrix plane represented plane that maps plane with respect Proof Prove quadratic form real matrices real numbers real symmetric matrix reflection matrices representing a rotation rotation matrix row transformation matrices scalar multiple set of points shear parallel similar matrices single-valued mapping skew-Hermitian matrix skew-symmetric matrix square matrix symmetric matrix system of linear transformation represented translation transpose x-axis x₁ y-axis y₁ Y₁)²