Week 1:
Sep 20-24
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First week of class. Primarily review of basic linear algebra concepts. Roughly Chap. 1 of Trefethen-Bau for more detail see Chaps. 1-3 of Strang.
- Monday:
- First day. Notation. Review basic algebraic properties. Discuss matrices as linear transformations.
- Wednesday:
- Review of complex numbers. Algebra of complex numbers. Euler's formula. Statement of Fundamental theorem of algebra. Interpretation of complex numbers in terms of linear algebra of $\mathbb{R}^2$. See Chap. 10 of Strang.
- Friday:
- Discuss span and linear independence of a set of vectors. Range and Null space of a $m\times n$ matrix defined. Subspaces of $\mathbb{C}^n$ and its basis. Begin discussion of Gaussian elimination.
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Week 2:
Sep 27-Oct 1
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Finish the review from week 1.
- Monday:
- Finish discussion of Gaussian elimination. Prove some basic facts.
- Wednesday:
- Finish proofs from Monday.
- Friday:
- The inverse of a matrix. Systems of equations.
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Week 3:
Oct 4-8
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Start talking about inner products for vectors and associated properties for matrices. This is Chaps. 2 and 6 of T-B
- Monday:
- Finish discussion of non-singular matrices. Start inner products and discussion of adjoint.
- Wednesday:
- First Homework Due. Continue discussion of innerproducts. Unitary matrices.
- Friday:
- Complementary subspaces. Projectors.
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Week 4:
Oct 11-15
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Begin disucssion of the QR factorization. This is Chap. 6 and Chap. 3 of Strang.
- Monday:
- Orthogonal projectors.
- Wednesday:
- Second Homework Due. Begin four fundamental spaces of a matrix.
- Friday:
- Finish four fundamental spaces of a matrix.
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Week 5:
Oct 18-22
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We discuss the QR factorization and give applications. This is Chaps. 7, 8 and 11.
- Monday:
- Gram-Schmidt orthogonalization. Begin QR factorization.
- Wednesday:
- Third Homework Due. Continue with QR factorization and give linear algebra application.
- Friday:
- Least Squares Problems.
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Week 6:
Oct 25-29
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This week we will finish discussion of QR factorization and have the midterm. We discuss preliminaries needed for he Singular Value Decomposition. This is Chaps. 3, 4 and 10
- Monday:
- Computing the QR factorization. Householder Triangularization.
- Wednesday:
- Midterm.
- Friday:
- Finish Householder Triangularization. Start Norms of Vectors and Norms of Matrices.
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Week 7:
Nov 1-5
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Begin discussion of SVD. This is Chap. 3-5 in T-B.
- Monday:
- Continue Norms of Vectors and Norms of Matrices.
- Wednesday:
- Fourth Homework Due. Begin Singular Value Decomposition (SVD).
- Friday:
- Geometry of Singular Value Decomposition and statement.
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Week 8:
Nov 8-12
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More on SVD. Begin Eigenvalues and Eigenvectors. Subset of Chaps. 24-31 of T-B.
- Monday:
- Proof of the existence of SVD. Uniqueness properties of SVD
- Wednesday:
- Sixth Homework Due. Applications of the SVD.
- Friday:
- Eigenvalues and Eigenvectors -- a whirlwind tour. Computing the SVD via Eigenmethods.
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Week 9:
Nov 15-19
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Eigenvalues and Eigenvectors. This is of Chap. 24 and Chap. 27 of T-B.
- Monday:
- More on computing the SVD. The Schur factorization.
- Wednesday:
- Fifth Homework Due. More on the Schur factorization. Applications of diagonalizable matrices.
- Friday:
- The Rayleigh quotient and power iteration method.
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Week 10:
Nov 22-26
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No class -- Thanksgiving
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Week 11:
Nov 29-Dec 3
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More advanced algorithms. For instance conjugate gradients. This is Chaps. 32 and 38 of T-B.
- Monday:
- Discussion of iterative methods -- in particular approachs to solving systems of equations. Krylov subspaces
- Wednesday:
- Method of conjugate gradients.
- Friday:
- Seventh Homework Due (Post-poned). Review Day.
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