Math 130 Linear Algebra
[This course page is under development.]
Please bookmark this page, http://aleph0.clarku.edu/~ma130/, so you can readily access it.
We won’t cover all of the topics listed below at the same depth. Some topics are fundamental and we’ll cover them in detail; others indicate further directions of study in linear algebra and we’ll treat them as surveys. Besides those topics listed below, we will discuss some applications of linear algebra to other parts of mathematics and statistics and to physical and social sciences.
Matrices and vectors
Matrix addition and scalar multiplication.
Matrix multiplication. Matrix algebra. Matrix inverses.
Powers of a matrix. The transpose and symmetric matrices.
Vectors in Rn.
Developing geometric insight.
Planes in R3.
Lines and hyperplanes in Rn.
Systems of linear equations
Systems of linear equations.
Homogeneous systems and null space.
Matrix inversion and determinants
Matrix inverse using row operations.
Determinants. Results on determinants.
Matrix inverse using cofactors.
Leontief input-output analysis.
Rank, range and linear equations
The rank of a matrix.
Rank and systems of linear equations.
Linear independence, bases and dimension
Basis and dimension in Rn.
Linear transformations and change of basis
Range and null space.
Change of basis and similarity
Eigenvalues and eigenvectors.
Diagonalisation of a square matrix.
When is diagonalisation possible?
Applications of diagonalisation
Powers of matrices.
Systems of difference equations.
Linear systems of differential equations.
Inner products and orthogonality
Gram-Schmidt orthonormalisation process
Orthogonal diagonalisation and its applications
Orthogonal diagonalisation of symmetric matrices.
Direct sums and projections
The direct sum of two subspaces.
Characterizing projections and orthogonal projections.
Orthogonal projection onto the range of a matrix. Minimizing the distance to a subspace.
Fitting functions to data: least squares approximation.
Complex matrices and vector spaces
Complex vector spaces.
Complex inner product spaces.
Unitary diagonalisation and normal matrices.
Class notes, quizzes, tests, homework assignments
- Vectors and vector spaces.
- Vectors in Rn.
Their addition, subtraction, and multiplication
by scalars (i.e. real numbers). Graphical interpretation of these vector operations
- Topics we’ll discuss at the end of the semester when we discuss inner product
spaces: norm of a vector (also called length), unit vectors,
inner products of vectors (also called dot products).
- Real vector spaces defined abstractly. Axioms and theorems that follow from them.
- Examples of vector spaces besides Rn. Matrices, row
vectors, column vectors, polynomials, infinite sequences
- Fields defined abstractly. Axioms and theorems that follow from them.
- The complex field C and complex vector spaces
- Finite fields and their vector spaces.
- Subspaces. Lines in the plane R2,
lines and planes in R3. The 0 subspace.
- Systems of simultaneous linear equations and their solutions. Row reduction.
Solving them with Matlab.
- Linear combinations of vectors, the span of a set of vectors.
Geometric interpretation of the span.
Span and linear combination problems in Matlab
- Linear dependence and independence.
Geometric interpretation of dependence and independence
Testing for linear independence in Matlab.
- Bases and dimension. The dimension of subspaces.
Finite-dimensional versus infinite-dimensional spaces.
- Linear Transformations and Matrices.
- Linear transformations. The definition of a linear transformation
L: V → W from the domain space V to
the codomain space W. When V = W, L is
also called a linear operator on V.
- Examples L:
Rn → Rm.
Linear operators on R2 including
rotations and reflections, dilations and contractions, shear transformations,
projections, the identity and zero transformations
- The null space (kernel) and the range (image) of a transformation, and their
dimensions, the nullity and rank of the transformation
- The dimension theorem: the rand plus nullity equals the dimension of the domain
- Matrix representation of a linear transformation between finite dimensional
vector spaces with specified bases
- Operations on linear transformations V → W. The
vector space of all linear transformations V → W.
Composition of linear transformations
- Corresponding matrix operations, in particular, matrix multiplication
corresponds to composition of linear transformations. Powers of square matrices.
Matrix operations in Matlab
- Invertibility and isomorphisms. Invariance of dimension under isomorphism.
- The change of coordinate matrix between two different bases of a vector space.
- Dual Spaces.
- [A matrix representation for complex numbers, and another for quaternions.
Historical note on quaternions.]
- Elementary matrix operations and systems of simultaneous linear equations.
- Elementary row operations and elementary matrices.
- The rank of a matrix (row rank) and of its dual (its column rank).
- An algorithm for inverting a matrix. Matrix inversion in Matlab
- Systems of linear equations in terms of matrices.
Coefficient matrix and augmented matrix.
Homogeneous and nonhomogeneous equations.
Solution space, consistency and inconsistency of systems.
- Reduced row-echelon form, the method of elimination (sometimes called
Gaussian elimination or Gauss-Jordan reduction)
- 2x2 Determinants of Order 2. Multilinearity. Inverse of a 2x2 matrix.
Signed area of a plane parallelogram, area of a triangle.
- nxn determinants. Cofactor expansion
- Computing determinants in Matlab
- Properties of determinants. Transposition, effect of elementary row
operations, multilinearity. Determinants of products, inverses, and transposes.
Cramer’s rule for solving n equations in n unknowns.
- Signed volume of a parallelepiped in 3-space
- [Optional topic: permutations and inversions of permutations; even and odd permutations]
- [Optional topic: cross products in R3]
- Eigenvalues and eigenvectors of linear operators
- An eigenspace of a linear operator is a subspace in which the operator acts
as multiplication by a constant, called the eigenvalue (also called the
characteristic value). The vectors in the
eigenspace are alled eigenvectors for that eigenvalue.
- Geometric interpretation of eigenvectors and eigenvalues.
Fixed points and the 1-eigenspace.
Projections and their 0-eigenspace.
Reflections have a –1-eigenspace.
- Diagonalization question.
- Characteristic polynomial.
- Complex eigenvalues and rotations.
- An algorithm for computing eigenvalues and eigenvectors
- Inner product spaces
- Inner products for real and complex vector spaces (for real vector spaces, inner
products are also called dot products or scalar products) and norms (also called
lengths or absolute values). Inner product spaces. Vectors in Matlab.
- The triangle inequality and the Cauchy-Schwarz inequality, other properties
of inner products
- The angle between two vectors
- Orthogonality of vectors ("orthogonal" and "normal" are other words for
- Unit vectors and standard unit vectors in Rn
- Orthonormal basis
To be filled in as the course progresses.
Some old linear algebra tests
Dave’s Short Course on Complex Numbers
- Writing Proofs: structures of theorems and proofs,
synthetic and analytic proofs, logical symbols, and well-written proofs
- Vector Spaces, their axiomatic definition
- Properties of vector spaces that follow from the axioms
- Subspaces of vector spaces
- Linear differential equations, an application of
- A little bit about sets
- Simultaneous systems of linear equations.
The Chinese method of elimination at
- Notes on Matlab
- Systems of linear equations, elementary row operations,
and reduced echelon form...
- Brief introduction to Latex and its
- Linear combinations and basis
- Span, and independence
- The replacement theorem and dimension
- Where dimension doesn’t work
- Linear transformations, matrices, and linear operators
- Some linear transformations of the plane
- Composition and categories, composition of linear transformations and multiplication of matrices
- Algebra of linear transformations and matrices
- Kernel, image, nullity, and rank
- Inverses of linear transformations and matrices
- Change of coordinates
- Elementary row operations and elementary matrices
- Rank and nullity of matrices
- Introduction to determinants, 2x2 and 3x3 determinants, areas of triangles and parallelagrams in the plane, volumes of parallelepipeds, Jacobians
- Permutations and determinants
- Characterizing properties and constructions of determinants, cofactors, diagonal and triangular matrices
- More properties of determinants, an algorithm for evaluating determinants, determinants of products, inverses, and transposes, Cramer’s rule
- Eigenvalues, eigenvectors, and eigenspaces of linear operators
- Rotations and complex eigenvalues
- Diagonalizable operators and matrices
- Norm and inner products in Rn
- Applications of inner products in Rn
- Norm and inner products in Cn
and abstract inner product spaces
- Cauchy’s inequality
- Cross products
Web pages for related courses
Pages on the web that you may find interesting
This page is located on the web at