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Welcome to the Linear Algebra Wiki[]

Learn about linear algebra

Syllabus[]

1a[]

Vector quantities and $R^n$

$R^{2}$ and analytic geometry.

Points, line segments and lines. Parametric vector equations. Parallel lines.

Planes. Linear combinations and the span of two vectors. Planes though the origin. Parametric vector equations for planes in Rn: The linear equation form of a plane.

Use of vectors to prove geometric theorems; parametric vector equations for rays, line segments, parallelograms, triangles; elements of vector calculus.

Linear Equations and Matrices

Introduction to systems of linear equations. Solution of 2x2 and 2x3 systems and geometrical interpretations.

Matrix notation. Elementary row operations.

Solving systems of equations via Gaussian elimination.

Deducing solubility from row-echelon form. Solving systems with indeterminate right hand side.

General properties of solutions to $Ax=b$ . Applications. or Matrix operations

Elementary matrices and elementary row operations, applications of linear equations and matrices to electrical engineering (Kirchhoff's Laws), economics (Leontief model).

Matrices

Operations on matrices. Transposes.

Inverses and definition of determinants.

Properties of determinants.

Vector Geometry

Length, angles and dot product in $R^{2}$ , $R^3$ , $Rn$ .

Orthogonality and orthonormal basis, projection of one vector on another. Orthonormal basis vectors. Distance of a point to a line.

Cross product: definition and arithmetic properties, geometric interpretation of cross product as perpendicular vector and area.

Scalar triple products, determinants and volumes . Equations of planes in $R^3$ : the parametric vector form, linear equation (Cartesian) form and point-normal form of equations, the geometric interpretations of the forms and conversions from one form to another. Distance of a point to a plane in $R^3$ .

Use of vectors to prove geometric theorems, further applications of vectors to physics and engineering, rotations of Cartesian coordinate systems and orthogonal matrices

1b[]

Vector Spaces

introduce the general theory of vector spaces and to give some basic examples. The majority of examples will be for the real vector space Rn, but occasional examples may be given for the complex vector space Cn, as well as from vector spaces of polynomials.

Introduction to vector spaces and examples of vector spaces

Properties of vector arithmetic

Subspaces.

Linear combinations and spans.

Linear independence.

Basis and dimension .

Introduction to linear maps. Linear maps and the matrix equation .

Geometrical examples .

Subspaces associated with linear maps .

Rank, nullity and solutions of Ax = b. Further applications .

Eigenvalues and Eigenvectors

Definition, examples and geometric interpretation of eigenvalues and eigenvectors.

Eigenvectors, bases and diagonalization of matrices .

Applications to powers of matrices and solution of systems of linear differential equations .

1b higher:

Vector spaces. Matrices, polynomials and real-valued functions as vector spaces . Coordinate vectors .

Linear transformations.Linear maps between polynomial and real-valued function vector spaces. Matrix representations for non-standard bases in domain and codomain .

Matrix arithmetic and linear maps. Injective, surjective and bijective linear maps.

Proof the rank-nullity theorem.

Eigenvalues and eigenvectors. Markov Chain Processes. Eigenvalues and eigenvectors for symmetric matrices and applications to conic sections.

2a[]

Basis

Vector spaces, linear transformations, change of basis, inner products, orthogonalization, least squares approximation, QR factorization , determinants, eigenvalues and eigenvectors, diagonalization, Jordan forms, matrix exponentials and applications to systems of differential equations, other applications of linear algebra

1: Linear Equations and Matrices
Gaussian elimination, back substitution.
Conditions for solubility.
Operations on matrices. Inverses.

2: Vector Spaces
Definition and examples. Subspaces.
Linear combinations, spans, linear independence.
Bases, dimension, coordinates.
Kernel, column space , rank and nullity of a matrix.

3: Linear Transformations
Linear transformations and matrix representations.
Change of basis.
Kernel, image, rank and nullity of a linear transformation.

4: Gram-Schmidt, Least Squares, QR Factorisation
Dot products in Rn and Cn.
Orthogonal and orthonormal sets.

Symmetric, skew-symmetric and orthogonal matrices.
Hermitian, skew-Hermitian and unitary matrices.
Gram-Schmidt orthogonalisation process.
QR factorisation by Gram-Schmidt.
Orthogonal components. Projections.
Least squares approximation.
Reflection in a hyperplane.

5: Determinants
Definition, computation and properties.

6: Eigenvalues, Eigenvectors and Diagonalisation
Eigenvectors, eigenvalues and eigenspaces.
Geometric and algebraic multiplicity.
Diagonalisation.
Relation of trace and determinant to eigenvalues.
Similarity of matrices. Similarity variants.

7: Symmetric and Hermitian matrices
Properties of symmetric and Hermitian matrices.
Diagonalisability.
Quadratic forms and representation by symmetric matrices.
Canonical forms for conics and quadratics.
Positive definite matrices .

8: Jordan Form
Direct sums of matrices. Jordan forms .
Deducing Jordan form from the nullities of (A- I)k.
Generalised eigenvectors.
Achieving Jordan form by change of basis.

9: Matrix Exponentials
Definition of exp(A) as a power series.
Action of exp(tA) on generalised eigenvectors of A.
Methods for finding exp(tA).

10: Systems of Ordinary Differential Equations
Homogeneous systems: the solution space, fundamental matrices, methods of solution.
Inhomogeneous systems: the method of variation of parameters.