EM 128 - LINEAR ALGEBRA AND ANALYTIC GEOMETRY II (Álgebra Linear e Geometria Analítica II) Linear Spaces. Definition and examples. Subspaces of a linear space. Dependent and independent sets in a linear space. Bases and dimension. Inner products. Euclidean spaces. Norms and orthogonality. Linear Transformations. Definition. Null space and range. Nullity and rank. Inverses. One-to-one linear transformations. Matrix representations of linear transformations. Eigenvalues and Eigenvectors. Linear transformations with diagonal matrix representations. Eigenvalues and eigenvectos of a linear transformation. Linear independence of eigenvectors corresponding to distinct eigenvalues. The finite-dimensional case. Characteristic polynomials. Calculation of eigenvector and eigenvalues in the finite-dimensonal case. Matrices representing the same linear transformatiom. Similar matrices. Eigenvalues in Euclidean spaces. Eigenvalues and inner products. Orthogonality of eigenvectors corresponding to distinct eigenvalues. Unitary matrices. Orthogonal matrices. Quadratic forms. Reduction of a real quadratic form to a diagonal form. Applications to analytic geometry. Vector-valued functions. Algebraic operations. Derivatives and integrals. Applications to curves. Tangency. Applications to curvilinear motion. The unit tangent, the principal normal and bi-normal vectors. Osculating, tangent and rectifying planes. Definition of arc length. The arc length function. Curvature of a curve. Torsion of a curve. Frenet- Serret formulas. <<<<<<<<<<  BACK  <<<<<<<<<<