18.06 Spring 2008Outline for Final Exam1. Elimination and solving linear systems- how to find particular and complete solutions- solvability, uniqueness, rank- Cramer’s rule2. Inverses- how to find them and use them- relationship to cofactors3. A = LU and PA=LU decompositions- row reduced echelon form R4. Vector spaces and subspaces- definitions, examples5. Linear independence and bases- span and dimension6. Linear transformations T- finding T(x) for x expressed in a basis- how to translate into a matrix7. Four subspaces- dimensions- how to find a basis for each- orthogonality properties8. Orthogonality9. Projection matrices- how to construct them- what they do- application to solve least squares10. Orthogonal matrices- basic properties11. Gram-Schmidt- how to do the Gram-Schmidt process- A = QR decomposition112. Determinants- definitions and properties- specific examples- methods: elimination, big formula, cofactors13. Eigenvalues and eigenvectors- how to find them- relationship to determinant and trace- examples and properties14. Diagonalization- how to find it- how to use it- solving differential equations15. Spectral theorem for symmetric matrices16. Positive definite matrices- properties and tests- why they are important- minimizing a quadratic17. Similarity- definition- relationship to diagonalization- Jordan canonical form18. Singular value decomposition- how to find it from A’A- what information it gives you19. Graphs and networks- translating graph questions into linear algebra- application to circuits20. Markov matrices- steady state and applications21. Complex matrices- complex dot products- complex analogues of symmetric, orthogonal, etc.Special matrices: pe rmutation, projection, rank one, sym metric, orthog-onal,
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