MA 54-02, Spring 2004
Student's notes.

• Jan. 28: Introduction. Vector spaces: definition and examples (PDF, 69K)
• Jan. 30: Bases. Generating and linearly independent systems of vectors. (PDF, 72K)
• Feb. 2: Linear transformations and matrix-vector multiplication (PDF, 105K)
• Feb. 4: Composition of linear transformations and matrix multiplication (PDF, 110K)
• Feb. 6: Invertible transformations and matrices. Isomorphisms (PDF, 117K)
• Feb. 9:  Applications to the computer graphics (PDF, 68K)
• Feb. 11: Many faces of linear systems. Row reduction (PDF, 64K)
• Feb. 13: Analyzing the pivots (PDF, 99K)
• Feb. 16: Subspaces, fundamental subspaces, structure of the general solution of a linear system (PDF, 57K)
• Feb. 18: Computing fundamental subspaces. Rank Theorem. (PDF, 64K)
• Feb 20: How to write proofs. Review of dimension and rank (PDF, 78 K)
• Feb. 25: Change of basis formula (PDF, 112 K)
• Feb. 27: Determinants: Introduction, properties.  (PDF, 56K)
• March 1: Determinants. Cofactor (row or column) expansion (PDF, 63K)
• March 3: Cofactor expansion, cofactor formula for the inverse (PDF, 146K)
• March 5: Formal definition of determinant. Permutations (PDF, 201 K)
• March 8: Review for the test
• March 10: Test 1
• March 12: Spectral Theory, main definitions: eigenvalues and eigenvectors. (PDF, 84K)
• March 15: Diagonalization (PDF, 121 K)
• March 17: Bases of subspaces. Proof of necessary and sufficient condition of diagonalizability (PDF, 59K)
• March 19: Inner product spaces. Introduction (PDF, 53K)
• March 22: Cauchy-Schwarz inequality (PDF, 56K)
• March 24: Orthogonal projection (PDF, 66K)
• March 26: Gram-Schmidt orthogonalization
• April 5: Least square solution (PDF, 107 K)
• April 7: Adjont operators. Fundamental subspaces revisited. (PDF, 59 K)
• April 9: Isometries and unitary operators. Unitarily equivalent operators. (PDF, 93 K).
• April 12: Schur (upper triangular) representation of a matrix. Spectral theorem for self-adjoint (Hermitian) matrices. (PDF, 53K)
• April 14: Normal operators. Diagonalization of normal operators.
• April 16. Positive definite (semi-definite) operators. Square roots.
• April 19: Review for the test 2
• April 21: Test 2
• April 23: Singular value decomposition
• April 26: Singular value decomposition, continued.
• April 28: What singular value decomposition tells us about?
• April 30: Bilinear and quadratic forms. Diagonalization of quadratic forms.
• May 2: Silvester's law of inertia. Positive definite forms. Sivester's criterion of positivity.