Linear Algebra
Linear systems, matrices, determinants, vector spaces, transformations and eigenvalues.
Textbook: Elementary Linear Algebra, Howard Anton, Wiley Publishing Co. (2000)
- Week 1- Introduction to Systems of Linear Equations
- Systems of linear equations: Gauss and Gauss-Jordan eliminationYou learn to turn a linear system into an augmented matrix and reduce it to REF and RREF with row operations, to tell a unique solution, infinitely many solutions and inconsistency apart, and to solve systems with parameters. The biggest block of the midterm.
- Week 2- Operations with Matrices
- Matrix operations and their propertiesYou learn matrix addition, scalar multiplication, matrix product, transpose and trace; when a product is defined; and the rules that behave differently from numbers, such as AB ≠ BA.
- Week 3- The Inverse of a Matrix
- Inverse matrix, the [A | I] method and elementary matricesYou learn to find the inverse of a matrix with the 2×2 formula and with Gauss-Jordan on [A | I], to recognize a singular matrix, to solve AX = B and XA = B from the correct side, and to write row operations as products with elementary matrices.
- Week 4- The Determinant of a Matrix
- Determinant: cofactor expansion and computing with row operationsYou learn the minor and cofactor, cofactor expansion along any row or column, the triangular matrix rule, and how row operations change the determinant, so you can compute 3×3 and 4×4 determinants without a calculator.
- Week 5- Properties of Determinants
- Determinant properties, the adjoint and Cramer's ruleYou learn the rules for det(AB), det(Aᵀ), det(cA), det(A⁻¹), the link between invertibility and the determinant, finding the inverse with the adjoint (adjugate), and solving systems with Cramer's rule.
- Week 6- Vectors in R^n
- Vectors in Rⁿ and vector spacesYou learn vector operations in Rⁿ, linear combinations, length and the dot product, then the ten axioms of a vector space and how to check whether a set (with standard or nonstandard operations) is a vector space.
- Week 7- Subspaces of Vector Spaces
- Subspaces and the subspace testYou learn to check with a three-step test whether a set inside a vector space is a subspace; to recognize the typical subspaces and trap sets in Rⁿ, M₂₂ and P₂; and how intersection and union behave.
- Span and linear independenceYou learn to test the subspace spanned by a set of vectors, whether the set spans the whole space, and whether the vectors are linearly independent, using a homogeneous system or a determinant, and to apply the same method in Rⁿ, P₂ and M₂₂.
- Week 8- Basis and Dimension
- Basis, dimension and the coordinate vectorYou learn to test whether a set is a basis, to find a basis and the dimension of a subspace given by equations, to extract a basis from a set, and to compute the coordinates of a vector relative to a basis.
- Week 9- Rank of a Matrix and Systems of Linear Equations
- Rank, row/column/null space and linear systemsYou learn to find bases for the row space, column space and null space of a matrix, rank and nullity, the rank-nullity theorem, and to decide the consistency of a system with rank and write its solution in the form 'particular solution + homogeneous solution'.
- Week 10- Introduction to Linear Transformations
- Introduction to linear transformationsYou learn to test with two conditions whether a function is a linear transformation, matrix transformations, projection, reflection and rotation in the plane, abstract examples such as the derivative and trace, and how to compute a linear transformation from the images of a basis.
- Week 11- The Kernel and Range of a Linear Transformation
- Kernel and range of a linear transformationYou learn to find the kernel and the range of a linear transformation, to compute their bases and dimensions (nullity, rank), the dimension theorem, and to test whether a transformation is one-to-one, onto or an isomorphism.
- Week 12- Matrices for Linear Transformations
- Transformation matrices, transition matrices and similarityYou learn to find the standard matrix of a linear transformation, the matrix of a composition and of an inverse transformation, the matrix of a transformation relative to nonstandard bases, the transition matrix between two bases with the [B' | B] method, and similar matrices (A' = P⁻¹AP).
- Week 13- Eigenvalues and Eigenvectors
- Eigenvalues and eigenvectorsYou learn to set up the characteristic equation and find eigenvalues, to compute a basis of the eigenspace for each eigenvalue, to tell algebraic and geometric multiplicity apart, and the shortcuts relating eigenvalues to trace, determinant, powers and inverse.
- Diagonalization and matrix powersYou learn to test whether a matrix is diagonalizable by counting eigenvectors, to build P and D and write A = PDP⁻¹, and to use it to compute high powers of A quickly.
- Week 14- Symmetric Matrices and Orthogonal Diagonalization
- Symmetric matrices and orthogonal diagonalizationYou learn orthogonal matrices (Pᵀ = P⁻¹), the special properties of symmetric matrices (real eigenvalues, orthogonal eigenspaces, always diagonalizable), how to diagonalize a symmetric matrix with an orthogonal P so that PᵀAP = D, and how to use Gram-Schmidt for a repeated eigenvalue.