Math 2568 - Autumn 2022

• Lecture 1 (Course overview and § 1.1: Introduction to Matrices And Systems of Linear Equations).
• Lecture 2 (§ 1.1 (cont.): Introduction to Matrices And Systems of Linear Equations, and § 1.2: Echelon Form and Gauss-Jordan Elimination).
• Lecture 3 (§ 1.2 (cont.): Gauss-Jordan Elimination).
• Lecture 4 (§ 1.3: Consistent Systems of Linear Equations).
• Lecture 5 (§ 1.3 (cont.): Consequences of Gauss-Jorgand, and § 1.5 Matrix operations).
• Lecture 6 (§ 1.6: Algebraic Properties of Matrix Operations).
• Lecture 7 (§ 1.6 (cont.): Algebraic Properties of Matrix Operations; The Euclidean space Rⁿ).
• Lecture 8 (§ 1.9: Matrix Inverses and Their Properties).
• Lecture 9 (§ 1.7: Singular Matrices and Linear independence).
• Lecture 10 (§ 2.1: Vectors in the Plane, and § 2.2: Vectors in Space).
• Lecture 11 (§ 2.3: The Dot Product).
• Lecture 12 (§ 2.3: The Cross Product).
• Lecture 13 (§ 2.4: Lines in the Plane; lines and planes in Space).
• Lecture 14 (§ 2.4 (cont.): Planes in Space; § 3.1: Introduction to the Vector Space Rⁿ, and § 3.2: Vector Space Properties of Rⁿ).
• Lecture 15 ( § 3.2 (cont.): Subspaces of Rⁿ, and § 3.3: Examples of Subspaces).
• Lecture 16 (§ 3.3 (cont.): Examples of Subspaces).
• Lecture 17 (§ 3.4: Bases for Subspaces).
• Lecture 18 (§ 3.4 (cont.): Bases for Subspaces, and § 3.5: Dimension of subspaces of Rⁿ).
• Lecture 19 (§ 3.4 (cont.): The rank-nullity theorem, and § 3.6: Orthogonal Bases for Subspaces).
• Lecture 20 (§ 3.6: The Gram-Schmidt algorithm).
• Lecture 21 (§ 3.7: Linear Transformation from Rⁿ to R^m).
• Lecture 22 (§ 3.7 (cont.): Linear Transformation from Rⁿ to R^m).
• Lecture 23 (§ 3.7 (cont.): The rank and nullity of linear transformations, § 5.1: Introduction to Vector Spaces and Linear Transformations, and § 5.2: Abstract Vector Spaces).
• Lecture 24 (§ 5.1-5.2: Abstract Vector Spaces, § 5.3: Subspaces).
• Lecture 25 (§ 5.4: Linear Independence and Bases).
• Lecture 26 (§ 5.4 (cont.): Coordinates relative to bases, and § 5.7: Linear Transformations).
• Lecture 27 (§ 5.7 (cont.): the Nullspace and range of a linear transformation, key example: coordinates relative to a basis).
• Lecture 28 (§ 5.7 (cont.): One-to-one, onto and invertible linear transformations, § 5.8: Operations With Linear Transformations: addition, scalar multiplication, composition).
• Lecture 29 (§ 5.9: Matrix Representations Of Linear Transformations).
• Lecture 30 (§ 6.1: Introduction to Determinants and § 6.2: Cofactor Expansions Of Determinants).
• Lecture 31 (§ 6.3: Elementary Operations And Determinants).
• Lecture 32 (§ 4.1: The Eigenvalue Problem for (2 × 2) Matrices, and § 4.2: Determinants and the Eigenvalue Problem, § 4.4: Eigenvalues and the Characteristic Polynomial).
• Lecture 33 (§ 4.5: Eigenvectors and Eigenspaces; defective matrices).
• Lecture 34 (§ 4.7: Similarity Transformations and Diagonalization; real symmetric matrices).
• Lecture 35 (§ 4.6: Complex numbers and complex eigenvalues; roots of non-constant polynomials with real coefficients).
• Lecture 36 (§ 4.6: Complex Eigenvectors and Eigenspaces).

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