SCIENTIFIC COMPUTING - part 2 (EN - since A.Y. 2022-2023)
Indice degli argomenti
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Scientific Computing
part 2 - (6 credits)
prof. Mariarosaria Rizzardi
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Definitions and some examples in R2 and R3. Linear independence and basis of a space. Dimension of a Linear Space or Subspace, and related properties.The four fundamental subspaces of a matrix.
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Updated on 13/3/2024
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Definition and examples of a dot product between real vectors, and between complex vectors. Definition and examples of vector norms. Induced vector norms. Parallelogram Law. Induced matrix norms. Geometrical interpretation of matrix norms. Condition number of a matrix. Orthogonality and linear independence. Another definition of the standard scalar product. Applications of dot products and norms.
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"Sum" and "Direct Sum" subspaces; "Complementary subspaces" and "Orthogonal Complement" of a subspace. Grassmann Formula and other related properties. Intersection of two linear subspaces. Fundamental Theorem of Linear Algebra (about the 4 fundamental subspaces associated to a matrix). Gram-Schmidt Orthonormalization algorithm. Its connection to the QR factorization of a matrix.
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Affine spaces and subspaces, and related properties. Parallelism between affine subspaces. Intersection between affine subspaces. Affinely independent points and affine reference systems of subspaces.
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Linear Transformations (homomorphisms) and some properties. Definitions of isomorphism, endomorphism, automorphism. Kernel and Range of a linear map. Injective and surjective linear maps. Theor.: R(AT) and R(A) are isomorphic. An automorphism as a change of basis; advantage in using an orthonormal basis. Examples of 2D and 3D elementary linear maps. Factorization of a 2D tA into elementary linear maps. Some of orthogonal matrix properties. Generalized inverses of a matrix, ABCD Theorem, pseudoinverses and one sided inverses. Solutions of an undetermined linear system. Least-norm solution of an undetermined linear system.
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Real affine transformations and their Representation Theorem. Affinities. Examples. Main properties. How to detect an affine map given N points in the domain Rn and their images in the codomain Rm: condition to assure its uniqueness. Elementary 2D and 3D real affine maps. Translation matrix in homogeneous coordinates. Advantage in using homogeneous coordinates: comparison between the computational complexity of a roto-translation of N points in cartesian coordinates and in homogeneous coordinates. The MATLAB patch() function to draw 3D objects.
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Eigenvalues and eigenvectors. Computing the eigenvalues. Spectrum, Spectral Radius, Gershgorin’s Circle Theorem for locating eigenvalues. Eigenspaces. Computing the eigenvectors. Algebraic and geometric multiplicity of an eigenvalue. Properties of eigenvalues/eigenvectors of a generic square matrix, and of a symmetric matrix. Geometrical interpretation of eigenvalues/eigenvectors. Eigenvalues/eigenvectors of particular symmetric matrices. Eigenvalues/Eigenvectors of some elementary linear maps. Diagonalization of a matrix. Diagonalization of a matrix. Spectral Theorem for a symmetric matrix. Connection between SVD and diagonalization. Examples of applications of diagonalization and of eigenvalues/eigenvectors.
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Outlines of inferential statistics (Data matrix. Sample mean. Centered data matrix and standardized matrix. Scatter matrix. Covariance matrix and Correlation matrix): from a statistical point of view, and from a Linear Algebra point of view. PCA idea. Incremental PCA Algorithm derivation. PCA algorithms. Geometrical interpretation of PCA. PCA application: "Eigenfaces" algorithm for facial recognition. Regression in 2D and in 3D. PCA vs Least Squares.
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