Math 220: Linear Algebra
Spring 2023
- Syllabi and Calendar
- Notes and videos
- Assessments
- Discussion seminars, readings, videos, etc.
- Shared Documents
- ***Other
About Linear Algebra
- A brief history of linear algebra (here)
- Proof: The Language of Mathematics (here)
- Seeing the Light (A PowerPoint on using Linear Algebra for lighting graphics in 3D)
- Googling Markov (PDF showing how Google used Markov chains to generate pagerank)
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Topical Videos Videos by Topic (from the Khan Academy) Introduction to matrices Matrix multiplication (part 1) Matrix multiplication (part 2) Inverse Matrix (part 1) Inverting matrices (part 2) Inverting Matrices (part 3) Matrices to solve a system of equations Matrices to solve a vector combination problem Singular Matrices 3-variable linear equations (part 1) Solving 3 Equations with 3 Unknowns Linear Algebra: Introduction to Vectors Linear Algebra: Vector Examples Linear Algebra: Parametric Representations of Lines Linear Combinations and Span Linear Algebra: Introduction to Linear Independence More on linear independence Span and Linear Independence Example Linear Subspaces Linear Algebra: Basis of a Subspace Vector Dot Product and Vector Length Proving Vector Dot Product Properties Proof of the Cauchy-Schwarz Inequality Linear Algebra: Vector Triangle Inequality Defining the angle between vectors Defining a plane in R3 with a point and normal vector Linear Algebra: Cross Product Introduction Proof: Relationship between cross product and sin of angle Dot and Cross Product Comparison/Intuition Matrices: Reduced Row Echelon Form 1 Matrices: Reduced Row Echelon Form 2 Matrices: Reduced Row Echelon Form 3 Matrix Vector Products Introduction to the Null Space of a Matrix Null Space 2: Calculating the null space of a matrix Null Space 3: Relation to Linear Independence Column Space of a Matrix Null Space and Column Space Basis Visualizing a Column Space as a Plane in R3 Proof: Any subspace basis has same number of elements Dimension of the Null Space or Nullity Dimension of the Column Space or Rank Showing relation between basis cols and pivot cols Showing that the candidate basis does span C(A) A more formal understanding of functions Vector Transformations Linear Transformations Matrix Vector Products as Linear Transformations Linear Transformations as Matrix Vector Products Image of a subset under a transformation im(T): Image of a Transformation Preimage of a set Preimage and Kernel Example Sums and Scalar Multiples of Linear Transformations More on Matrix Addition and Scalar Multiplication Linear Transformation Examples: Scaling and Reflections Linear Transformation Examples: Rotations in R2 Rotation in R3 around the X-axis Unit Vectors Introduction to Projections Expressing a Projection on to a line as a Matrix Vector prod Compositions of Linear Transformations 1 Compositions of Linear Transformations 2 Linear Algebra: Matrix Product Examples Matrix Product Associativity Distributive Property of Matrix Products Linear Algebra: Introduction to the inverse of a function Proof: Invertibility implies a unique solution to f(x)=y Surjective (onto) and Injective (one-to-one) functions Relating invertibility to being onto and one-to-one Determining whether a transformation is onto Exploring the solution set of Ax=b Matrix condition for one-to-one trans Simplifying conditions for invertibility Showing that Inverses are Linear Deriving a method for determining inverses Example of Finding Matrix Inverse Formula for 2x2 inverse 3x3 Determinant nxn Determinant Determinants along other rows/cols Rule of Sarrus of Determinants Determinant when row multiplied by scalar (correction) scalar muliplication of row Determinant when row is added Duplicate Row Determinant Determinant after row operations Upper Triangular Determinant Simpler 4x4 determinant Determinant and area of a parallelogram Determinant as Scaling Factor Transpose of a Matrix Determinant of Transpose Transposes of sums and inverses Transpose of a Vector Rowspace and Left Nullspace Visualizations of Left Nullspace and Rowspace Orthogonal Complements Rank(A) = Rank(transpose of A) dim(V) + dim(orthogonoal complelent of V)=n Representing vectors in Rn using subspace members Orthogonal Complement of the Orthogonal Complement Orthogonal Complement of the Nullspace Unique rowspace solution to Ax=b Rowspace Solution to Ax=b example Showing that A-transpose x A is invertible Projections onto Subspaces Visualizing a projection onto a plane A Projection onto a Subspace is a Linear Transforma Subspace Projection Matrix Example Another Example of a Projection Matrix Projection is closest vector in subspace Least Squares Approximation Least Squares Examples Another Least Squares Example Linear Algebra: Coordinates with Respect to a Basis Change of Basis Matrix Invertible Change of Basis Matrix Transformation Matrix with Respect to a Basis Alternate Basis Tranformation Matrix Example Alternate Basis Tranformation Matrix Example Part 2 Changing coordinate systems to help find a transformation matrix Introduction to Orthonormal Bases Coordinates with respect to orthonormal bases Projections onto subspaces with orthonormal bases Finding projection onto subspace with orthonormal basis example Example using orthogonal change-of-basis matrix to find transformation matrix Orthogonal matrices preserve angles and lengths The Gram-Schmidt Process Gram-Schmidt Process Example Gram-Schmidt example with 3 basis vectors Introduction to Eigenvalues and Eigenvectors Proof of formula for determining Eigenvalues Example solving for the eigenvalues of a 2x2 matrix Finding Eigenvectors and Eigenspaces example Eigenvalues of a 3x3 matrix Eigenvectors and Eigenspaces for a 3x3 matrix Showing that an eigenbasis makes for good coordinate systems Vector Triple Product Expansion (very optional) Normal vector from plane equation Point distance to plane and Distance Between Planes |
Full Lectures Video Lectures from MIT OpenCourseWare 1. The Geometry of Linear Equations 2. Elimination with Matrices 3. Multiplication and Inverse Matrices 4. Factorization into A = LU 5. Transposes, Permutations, Spaces R^n 6. Column Space and Nullspace 7. Solving Ax = 0: Pivot Variables, Special Solutions 8. Solving Ax = b: Row Reduced Form R 9. Independence, Basis, and Dimension 10. The Four Fundamental Subspaces 11. Matrix Spaces; Rank 1; Small World Graphs 12. Graphs, Networks, Incidence Matrices 13. Quiz 1 Review 14. Orthogonal Vectors and Subspaces 15. Projections onto Subspaces 16. Projection Matrices and Least Squares 17. Orthogonal Matrices and Gram-Schmidt 18. Properties of Determinants 19. Determinant Formulas and Cofactors 20. Cramer's Rule, Inverse Matrix, and Volume 21. Eigenvalues and Eigenvectors 22. Diagonalization and Powers of A 23. Differential Equations and exp(At) 24. Markov Matrices; Fourier Series 24b. Quiz 2 Review 25. Symmetric Matrices and Positive Definiteness 26. Complex Matrices; Fast Fourier Transform 27. Positive Definite Matrices and Minima 28. Similar Matrices and Jordan Form 29. Singular Value Decomposition 30. Linear Transformations and Their Matrices 31. Change of Basis; Image Compression 32. Quiz 3 Review 33. Left and Right Inverses; Pseudoinverse 34. Final Course Review |