
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.
General information
Syllabus
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
Matrices.
Matrix addition and scalar multiplication.
Matrix multiplication. Matrix algebra. Matrix inverses.
Powers of a matrix. The transpose and symmetric matrices.
Vectors in R^{n}.
Developing geometric insight.
Lines.
Planes in R^{3}.
Lines and hyperplanes in R^{n}.
Systems of linear equations
Systems of linear equations.
Row operations.
Gaussian elimination.
Homogeneous systems and null space.
Matrix inversion and determinants
Matrix inverse using row operations.
Determinants. Results on determinants.
Matrix inverse using cofactors.
Leontief inputoutput analysis.
Rank, range and linear equations
The rank of a matrix.
Rank and systems of linear equations.
Range.
Vector spaces
Vector spaces.
Subspaces.
Linear span.
Linear independence, bases and dimension
Linear independence.
Bases. Coordinates.
Dimension.
Basis and dimension in R^{n}.
Linear transformations and change of basis
Linear transformations.
Range and null space.
Coordinate change.
Change of basis and similarity
Diagonalisation
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
Inner products.
Orthogonality.
Orthogonal matrices.
GramSchmidt orthonormalisation process
Orthogonal diagonalisation and its applications
Orthogonal diagonalisation of symmetric matrices.
Quadratic forms.
Direct sums and projections
The direct sum of two subspaces.
Orthogonal complements.
Projections.
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 numbers.
Complex vector spaces.
Complex matrices.
Complex inner product spaces.
Hermitian conjugates.
Unitary diagonalisation and normal matrices.
Spectral decomposition.
 Vectors and vector spaces.
 Vectors in R^{n}.
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 R^{n}. 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
C^{n}.
 Finite fields and their vector spaces.
 Subspaces. Lines in the plane R^{2},
lines and planes in R^{3}. 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.
Finitedimensional versus infinitedimensional 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:
R^{n} → R^{m}.
Linear operators on R^{2} 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.
Inverse matrices
 The change of coordinate matrix between two different bases of a vector space.
Similar matrices.
 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 rowechelon form, the method of elimination (sometimes called
Gaussian elimination or GaussJordan reduction)
 Determinants.
 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 3space
 [Optional topic: permutations and inversions of permutations; even and odd permutations]
 [Optional topic: cross products in R^{3}]
 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 1eigenspace.
Projections and their 0eigenspace.
Reflections have a –1eigenspace.
 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 CauchySchwarz inequality, other properties
of inner products
 The angle between two vectors
 Orthogonality of vectors ("orthogonal" and "normal" are other words for
"perpendicular")
 Unit vectors and standard unit vectors in R^{n}
 Orthonormal basis
Class notes, quizzes, tests, homework assignments
To be filled in as the course progresses.
 Vectors
 Fields

Dave’s Short Course on Complex Numbers
 Writing Proofs: structures of theorems and proofs,
synthetic and analytic proofs, logical symbols, and wellwritten 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
linear algebra
 A little bit about sets
 Simultaneous systems of linear equations.
The Chinese method of elimination at
http://aleph0.clarku.edu/~djoyce/ma105/simultaneous.html
 Notes on Matlab
 Systems of linear equations, elementary row operations,
and reduced echelon form...
 Brief introduction to Latex and its
source file
 Linear combinations and basis
 Span, and independence
 The replacement theorem and dimension
 Where dimension doesn’t work
 Isomorphisms
 Linear transformations, matrices, and linear operators
 Some linear transformations of the plane
R^{2}
 Coordinates
 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 R^{n}
 Applications of inner products in R^{n}
 Norm and inner products in C^{n}
and abstract inner product spaces
 Cauchy’s inequality
 Quaternions
 Cross products
Some old linear algebra tests
Web pages for related courses
Pages on the web that you may find interesting
This page is located on the web at
http://aleph0.clarku.edu/~ma130/