More during recitation + read the chapter in textbook for full background
- Vectors and matrices
- Introduce notations
- Subspace and linear independence
- Core concept in linear algebra
- Important when solving linear systems
- Dot product
- Shows up in many places.
- Example: vector norms, matrix-vector and matrix-matrix products.
- Used to define orthogonality
- Vector norms
- How to measure things
- Key to calculating errors in numerical methods
- Projection
- Using the dot product and norm for projection
- Pythagorean theorem
- How to simply calculate the length of a vector given its decomposition into orthogonal subspaces
- This is key to computing the norm of a vector in certain situations.
- Cauchy-Schwarz
- Key in proofs to derive upper bounds on error
- Considered one of the most important and widely used inequalities in mathematics
- Matrix-vector and matrix-matrix product
- Can be either viewed algebraically or interpreted as an operator
- Matrix-vector: applying a linear operator to transform a vector
- Matrix-matrix: operator composition
- Invertible matrix
- A requirement to solve a linear system and obtain a unique solution
- Sherman-Morrison-Woodbury formula
- How to solve a linear system when we make a small perturbation
- Operator and matrix norms
- Measuring the size of operators
- Key when deriving error bounds and for proof.
- The four fundamental spaces
- Understanding the structure of linear operators
- How they transform the input vector and map subspaces
- Orthogonal matrix and projector
- Orthogonal matrices will be key because they act as isometries
- Useful for building algorithms with low error
- Key in many matrix decompositions or factorizations
- Eigenvalues
- Key to analyzing the powers of a matrix, , and solving differential equations .
- This is important for time evolution and repeated applications of an operator.
- Long-term evolution of a dynamical system.
- Not useful to understand what happens when applying the operator once.
- Diagonalizable matrices
- A special matrix with a basis of eigenvectors.
- Determinant
- How a matrix changes the volume of a subspace.
- Trace
- Connect matrix with vector field
- Trace = divergence of vector field = flux through a unit square
- Unitarily diagonalizable matrices
- The simplest and most accurate case
- Diagonalizable + orthogonal matrices!
- Hermitian and symmetric matrices
- An important special case of unitarily diagonalizable matrices.
- Schur decomposition
- This will be key to computing eigenvalues.
- The fact that it uses orthogonal matrices will be important to ensure the accuracy of the algorithm.
- Exists for all square matrices.
- Singular value decomposition
- Key to understanding how a matrix scales and transforms space
- A linear operator always transforms the unit ball to an -ellipsoid.
- Connection with the four fundamental spaces.
- Connection with eigenvalues and determinant of a matrix.
- Key when solving least-squares problems: .
- Relation to operator and matrix norms.